Dominated Minimal Separators are Tame (Nearly All Others are Feral)
DDominated Minimal Separators are Tame(Nearly All Others are Feral)
Peter Gartland ∗ Daniel Lokshtanov ∗ July 20, 2020
Abstract
A vertex set S in a graph G is a minimal separator if there exist vertices u and v that are indistinct connected components of G − S , but in the same connected component of G − S (cid:48) forevery S (cid:48) ⊂ S . A class F of graphs is called tame if there exists a constant k so that every graphin F on n vertices contains at most O ( n k ) minimal separators. If there exists a constant k sothat every graph in F on n vertices contains at most O ( n k log n ) minimal separators the class is strongly-quasi-tame . If there exists a constant c > F contains n -vertex graphs withat least c n minimal separators for arbitrarily large n then F is called feral . The classificationof graph classes into tame or feral has numerous algorithmic consequences, and has recentlyreceived considerable attention.A key graph-theoretic object in the quest for such a classification is the notion of a k - creature .A k -creature consists of 4 disjoint vertex sets A, B, X = { x , . . . , x k } , Y = { y , . . . y k } such that:(a) A and B are connected, (b) there are no edges from A to Y ∪ B and no edges from B to X ∪ A , (c) A dominates X (every vertex in X has a neighbor in A ) and B dominates Y and(d) x i y j is an edge if and only if i = j . It is easy to verify that a k -creature contains at least 2 k minimal separators. On the other hand, in a recent manuscript [Abrishami et al., Arxiv 2020]conjecture that every hereditary class F that excludes k -creatures for some fixed constant k istame.Our main result is a proof of a weaker form of the conjecture of Abrishami et al. Moreconcretely, we prove that a hereditary class F is strongly quasi-tame if it excludes k -creatures forsome fixed constant k and additionally every minimal separator can be dominated by anotherfixed constant k (cid:48) number of vertices. The tools developed on the way lead to a number ofadditional results of independent interest. (i) We obtain a complete classification of all hereditary graph classes defined by a finite setof forbidden induced subgraphs into strongly quasi-tame or feral. This substantially generalizesa recent result of Milaniˇc and Pivaˇc [WG’19], who classified all hereditary graph classes definedby a finite set of forbidden induced subgraphs on at most 4 vertices into tame or feral. (ii)
Weshow that every hereditary class that excludes k -creatures and additionally excludes all cycles oflength at least c , for some constant c , is tame. This generalizes the result of [Chudnovsky et al.,Arxiv 2019] who obtained the same statement for c = 5. (iii) We show that every hereditaryclass that excludes k -creatures and additionally excludes a complete graph on c vertices forsome fixed constant c is tame. (iv) Finally we show that the domination requirement in ourmain result can not be dropped. Specifically we give an example of a feral family that excludes100-creatures, disproving the aforementioned conjecture of Abrishami et al. ∗ University of California, Santa Barbara, USA. Emails: [email protected] , [email protected] a r X i v : . [ c s . D M ] J u l Introduction
Let G be a graph and u and v be distinct vertices in G . A vertex set S is a u - v - separator if u and v are in distinct components of G − S . The set S is a minimal u - v - separator if S is a u - v -separator, but no proper subset of S is a u - v -separator. Finally, S is a minimal separator if S isa minimal u - v -separator for some pair of vertices u and v . Minimal separators have a tremendousrole in the design of graph algorithms, both directly, such as in the structural characterization ofchordal graphs [BLS99] but also indirectly in optimization algorithms for graph separation androuting problems (for example [Men27, Mar06, RS95]). The theory of potential maximal cliques,developed by Bouchitt´e and Todinca [BT01] implies that a several fundamental graph problems,such as computing the treewidth and minimum fill in of a graph G can be done in time polynomialin the number of vertices of G and the number of minimal separators in G . Lokshtanov [Lok10]showed that the same result holds for computing the tree-length of the graph G , while Fomin etal. [FTV15] proved a general result that showed that a whole class of problems (including e.g. maximum independent set and minimum feedback vertex set ) can be solved in time polynomial inthe number of vertices and minimal separators of the graph. All of these algorithms require a listof all the minimal separators of G to be provided as input. However, the listing algorithms forminimal separators of Kloks and Kratsch [KK94] or Berry et al. [BBC00] can be used to computesuch a list in time polynomial in the number of vertices times a factor linear in the number ofminimal separators of G .This brings to the forefront the main question asked in this paper - which classes of graphshave polynomially many minimal separators? We will say that a graph class F is tame if thereexists an integer k so that every graph in F on n vertices has at most O ( n k ) minimal separators.A number of important graph classes have been shown to be tame, such Chordal [BLS99] (andmore generally Weakly Chordal [BT01]), Permutation (and, more generally d -Trapezoid [Kra96]),Circular Arc [KKW98] and Polygon Circle graphs [Suc03]. Most of these results date back to thelate 1990s and early 2000s. Much more recently [ACD +
20, CTTV19, CPPT19, MP19], research hasstarted to focus on a more systematic classification of which graph classes are tame and whichare not. Indeed the term tame for graph classes with polynomially many minimal separators wasdefined by Milaniˇc and Pivaˇc [MP19], who classified all hereditary (closed under vertex deletion)classes defined by a set of forbidden induced subgraphs, all of which have at most 4 vertices, astame or not tame.Building on the terminology of Milaniˇc and Pivaˇc [MP19], we will say that a class of graphs F is quasi-tame if there exist constants k, c such that every n -vertex graph in the family containsat most O ( n k log c n ) minimal separators. Further, F is strongly quasi-tame if it is quasi-tame with c ≤
1. On the opposite side of the spectrum, we will say that F is feral if there exists a constant c such that for every N ≥ n ≥ N such that F contains an n -vertex graph with atleast c n minimal separators.Abrishami et al [ACD +
20] define a structure, called a k -creature, the presence of which appearsto control, to a large extent, whether a graph has many or few minimal separators. A k - creature in a graph G is a four-tuple ( A, B , X = { x , x , . . . , x k } , Y = { y , y , . . . , y k } ) of mutually disjointvertex subsets of V ( G ), satisfying the following conditions (see Figure 1).1. A and B are connected,2. A and Y ∪ B are anti-complete (i.e. no vertex in A is adjacent to a vertex in B ∪ Y ) B isanti-complete with X ∪ Y .3. A dominates X (every vertex in X has a neighbor in A ) and B dominates Y , and4. x i y j is an edge if and only if i = j . 1igure 1: A graph induced by the vertices of a k-creature. The blue edges indicate that x i ( y i ) mayor may not be neighbors with x j ( y j ) A graph G is k - creature-free if there does not exists a tuple of vertex sets of V ( G ) that forma k -creature. It is easy to see that a k -creature contains at least 2 k minimal separators (selectprecisely one of { x i , y i } for every i ≤ k ). Because deleting a vertex can not increase the number ofminimal separators, a graph G that contains a k -creature contains at least 2 k minimal separators.Thus, a graph family F that contains n -vertex graphs with k -creatures for arbitrarily large n andwith k = Ω( k ) is feral. For F to not be tame it is sufficient for k to grow super-logarithmicallywith n (i.e n = 2 o (log k ) ). A sort of converse to this observation was conjectured in [ACD + Conjecture 1. [ACD +
20] For every fixed natural number k , the family of graphs that are k -creature-free is tame. Even if Conjecture 1 were to be true, it would still not give a complete characterization ofhereditary graph classes into tame or non-tame. In particular Abrishami et al [ACD +
20] give anexample of a tame hereditary class F that contains arbitrarily large k -creatures. Their examplecan also be slightly modified to show that there exist hereditary families that are neither tame norferal. This makes it appear that, at least for hereditary classes in their full generality, the boundarybetween tame and non-tame graph classes is so “strange-looking” that a complete dichotomy maybe out of reach, and that we therefore have to settle for sufficient conditions for tameness / non-tameness, and possibly complete characterizations for more well-behaved sub-classes of hereditaryfamilies. For an example, Conjecture 1, if true, would have yielded a complete dichotomy into tameor feral for all classes of graphs closed under induced minors (i.e closed under vertex deletion andedge contraction).Unfortunately it turns out that Conjecture 1 is false . In particular we give (in Section 3)an example of a feral family F that excludes 100-creatures. The family F consists of all k - twistedladders (see Section 3). Our main result is nevertheless that Conjecture 1 is true “in spirit”, in thesense that for large classes of hereditary families, excluding k -creatures does imply few minimalseparators. To state Theorem 1 we need to define k - skinny ladders . A k - skinny-ladder is a graph G consisting of two paths P l = (cid:96) (cid:96) . . . (cid:96) k and P r = r r . . . r k and a set { s , s , . . . , s k } of verticessuch that for every i , s i is adjacent to (cid:96) i and r i and to no other vertices. Theorem 1.
For every natural number k , the family of graphs that are k -creature free and do notcontain a k -skinny-ladder as an induced minor is strongly-quasi-tame. Theorem 1 suggests that other counterexamples to Conjecture 1 should resemble the coun-terexample we provide in Section 3. We do not have any examples of classes that are strongly2uasi-tame according to Theorem 1, and conjecture that the statement of Theorem 1 remains trueeven if strongly quasi-tame is replaced by tame.Excluding the k -skinny ladder is closely tied to domination of minimal separators. A vertexset X dominates S of every vertex in S is either in X or has a neighbor in X . An importantingredient in the proof of Theorem 1 (see Lemma 15) is that for every k there exists a k (cid:48) so thatif G excludes k -creatures and k -skinny ladders as induced minors then every minimal separator S in G is dominated by a set X on at most k (cid:48) vertices. In fact, because a k -skinny ladder isitself 5-creature-free and contains a minimal separator (namely S ) which can not be dominated by k − F that exclude k -creatures, the presence or absence of k -skinny ladders precisely characterizes whether every minimal separator of every graph in F canbe dominated by a constant size set of vertices.To demonstrate the power of Theorem 1 we show that it gives, as a pretty direct consequence,a complete classification of all hereditary graph classes defined by a finite set of forbidden inducedsubgraphs into strongly quasi-tame or feral. Indeed, it is an easy exercise to show that if a family F is defined by a finite set of forbidden induced subgraphs and contains skinny k -ladders for arbitrarilylarge k , then there exists a constant f such that F either contains all f -subdivisions of 3-regulargraphs (an f - subdivision of G is the graph obtained from G by replacing each edge of G by a path on f + 1 edges) or all line graphs (see [Die12] for a definition) of f -subdivisions of 3-regular graphs. Inthis case F is feral. Therefore, Theorem 1 proves Conjecture 1 for hereditary graph classes definedby a finite set of forbidden induced subgraphs, albeit with strongly quasi-tame instead of tame. Weobtain Theorem 2 by extracting a small set of graphs that themselves contain large k -creatures,such that graphs that contain k -creatures contain one of them as an induced subgraph. We referto Figure 2 as well as to Section 2 for the definitions of the graphs used in Theorem 2 Theorem 2.
Let F be a graph family defined by a finite number of forbidden induced subgraphs. Ifthere exists a natural number k such that F forbids all k -theta, k -prism, k -pyramid, k -ladder-theta, k -ladder-prism, k -claw, and k -paw graphs, then F is strongly-quasi-tame. Otherwise F is feral. Note that some of the graphs of Figure 2 share a name with graphs that appear in the work ofAbrishami et al. [ACD + +
20] prove that the family of (what they define to be) theta-free, pyramid-free, prism-free,and turtle-free graphs is tame. We remark that our results are incomparable to theirs, in the sensethat there are classes of graphs whose tameness follows from their work, but not ours, and viceversa.Theorem 2 substantially generalizes the main result of Milaniˇc and Pivaˇc [MP19], who obtaineda complete classification into tame or feral of hereditary graph classes characterized by forbiddeninduced subgraphs on at most 4 vertices. The generalization comes at a price - as our upper boundson the number of minimal separators are quasi-polynomial instead of polynomial.Finally, we explore for which classes we are able to improve our quasi-polynomial upper boundsto polynomial ones. Here, again, domination plays a crucial role. We show that for every pair k , k (cid:48) of integers every class of graphs that excludes k -creatures and additionally has the property thatevery minimal separator S is dominated by a set X of size k (cid:48) disjoint from S is tame. We thenproceed to show that graphs that exclude k -creatures and all cycles of length at least t for somepositive integer t , leading to the following result. Theorem 3.
For every pair of natural numbers k and r , the family of graphs that are C ≥ k -free, k -theta-free, k -prism-free, and k -pyramid-free is tame. Note that for k -ladders we only depict one of the many possibilities of how the orderingof the neighborhoods of the a i ’s and b i ’s may be arranged in L and R . Dashed lines represent theoption of having an arbitrary length path or just an edge (except for k -claw and k -paw graphs whichthe dotted line is always a path of length k .) The blue lines used in the k -ladder-theta, k -ladder-prism, and k -ladder graphs represents the option of either having or not having that edge, but foreach vertex adjacent to more than one of the blue edges, at least one of those blue edges must belongto the graph. G is C ≥ r - free if it contains no induced cycles of length at least r . Theorem 3is optimal in the sense that k -theta, k -prism, and k -pyramid graphs have at least 2 k − minimalseparators and therefore can have exponentially many minimal separators. Further, it substantiallystrengthens the results of Chudnovsky et al. [CPPT19], who prove the same statement but only for r = 5.Finally we show that also graph classes that exclude k -creatures, k -skinny ladders, as well asat least one clique satisfy the property that every minimal separator S can be dominated by aconstant size set X disjoint from S . This implies that also this family of graphs is tame. Theorem 4.
For any fixed natural number k , the family of graphs that are k -creature-free, containno k -skinny-ladder as an induced minor, and contain no minimal separator that has a clique of size k is tame. Theorem 5.
Let F be a graph family defined by a finite number of forbidden induced subgraphs.If there exists a natural number k such that F forbids all k -clique, k -theta, k -ladder-theta, k -claw,and k -paw graphs then F is tame. Otherwise, F contains all cliques or F is feral. All graphs in this paper are assumed to be simple, undirected graphs. We denote the edge set of agraph G by E ( G ) and the vertex set of a graph by V ( G ). If v ∈ V ( G ), then we use N G [ v ] to denotethe closed neighborhood of v in the graph G , i.e. the set of all neighbors v has in G together with v itself. We use N G ( v ) to denote the set N G [ v ] − { v } . If X ⊆ V ( G ), then N G [ X ] = (cid:83) x ∈ X N G [ x ] and N G ( X ) = N G [ X ] − X . When the graph G is clear from the context, we will use N [ v ], N ( v ), N [ X ],and N ( X ). If X ⊂ V ( G ), then we use G [ X ] to denote the induced subgraph of G with vertex set X and G − X denotes G [ V ( G ) − X ].Given a graph G , a non-empty set S ⊂ G is called a minimal separator if there are at least twodistinct connected components L and R of G − S such that N G ( L ) = N G ( R ) = S . If u ∈ L and v ∈ R then we call S a minimal u - v - separator or a u, v - minimal separator . We say a component X of G − S is an S -full component if N G ( X ) = S . A family of graphs F is called tame if thereexists exists a constant c such that for all G ∈ F , G has at most | V ( G ) | c minimal separators. Afamily of graphs F is called strongly - quasi - tame if there exists exists a constant c such that for all G ∈ F , G has at most | V ( G ) | c log( | V ( G ) | ) minimal separators. A family of graphs F is called f eral if there exists exists a constant c > N there exists a G ∈ F ,such that | V ( G ) | = n > N and G has at least c n minimal separators.Given a path P = v , v , ..., v k we call v and v k the endpoints of P , and all other vertices of P are internal vertices of P . The length of a path is the number of vertices in the path. Givena graph G and a graph H , G is said to be H -free or G forbids H if G does not contains H as aninduced subgraph. We will sometimes talk about the induced minors of a graph so being H -freeshould not be confused with G not containing H as an induced minor. If G does not contain H asan induced minor then that is precisely what we will say, that G does not contain H as an inducedminor. If G is a family of graphs such that every G ∈ G is H -free, then G is said to be H -free orthat G forbids H . Similarly, given a graph G and a family of graph H , G is said to be H -free or G forbids H if G does not contain any H ∈ H as an induced subgraph. If G is a family of graphssuch that every G ∈ G is H -free, then G is said to be H -free or that G forbids H .Let F be a family of graphs. We say that F is a f amily of graphs def ined by a f inite numberof f orbidden induced subgraphs if there exists a finite set of graphs H such that G ∈ F if andonly if G does not contain an induced subgraph isomorphic to any graph in H . We say that H isa set of forbidden subgraphs that define F , and if H ∈ H we say that F explicitly forbids H .5iven a graph G let H and K be two subsets of V ( G ). We say that H is anti - complete with K or that H and K are anti - complete if H and K are disjoint and every vertex in H is non-adjacentto every vertex in K in G . We extend this definition in an obvious way to allow H (and possibly K ) to be a subgraph of G by saying H is anti-complete with K if V ( H ) is anti-complete with K ( V ( K ) if K is also a subgraph). A set X ⊂ V ( G ) is said to dominate a set Y ⊂ V ( G ) if for every y ∈ Y either y ∈ X or there is an x ∈ X such that yx ∈ E ( G ).Given a graph G and an edge uv ∈ E ( G ) we denote by G uv the graph that results from contracting the edge uv in G , so V ( G uv ) = ( V ( G ) − { u, v } ) ∪ { w } and for x, y ∈ V ( G ) − { u, v } , xy ∈ E ( G uv ) if and only if xy ∈ E ( G ) and for x ∈ V ( G ) − { u, v } , xw ∈ E ( G uv ) if and only if x isneighbors with u and/or v in G . Given an induced path P of G , we denote by G P the graph thatresults from contracting each edge of P one at a time. Note that the resulting graph is independentof the order the edges are contracted in.Given two anti-complete graphs A and B with a ∈ A and b ∈ B , we define an operation gluinga to b which produces a new graph C where V ( C ) = [( V ( A ) ∪ V ( B )) − { u, v } )] ∪ { w } and for x, y ∈ V ( C ) − { w } , xy ∈ E ( C ) if and only if xy ∈ E ( A ), or xy ∈ E ( B ) and for x ∈ V ( C ) − { w } , xw ∈ E ( G uv ) if and only if x is neighbors with a in A and/or b in B . This is the exact same graphthat would result in adding an edge between a and b and then contracting that edge. Given a graph G we call a tuple ( A, B , { x , x , . . . , x k } , { y , y , . . . , y k } ) of mutually disjoint vertexsubsets of V ( G ) a k - creature if the following conditions hold: (see Figure 1 for a depiction of thegraph a k -creature induces) • G [ A ] and G [ B ] are connected and A is anti-complete with B . • for i with 1 ≤ i ≤ k , x i y i ∈ E ( G ), x i has at least one neighbor in A and x i and is anti-completewith B , y i has at least one neighbor in B and y i is anti-complete with A . • for i, j with i (cid:54) = j and 1 ≤ i, j ≤ k x i y j / ∈ E ( G ).A graph is said to be k -creature free if there does not exists a tuple of vertex sets of V ( G ) thatform a k -creature.We call a graph G a k - skinny - ladder if the following conditions hold: • V ( G ) = { (cid:96) , (cid:96) , . . . , (cid:96) k , } ∪ { s , s , . . . , s k } ∪ { r , r , . . . , r k }• For all i with 1 ≤ i < k (cid:96) i (cid:96) i +1 ∈ E ( G ) and r i r i +1 ∈ E ( G ), for all i with 1 ≤ i ≤ k (cid:96) i s i ∈ E ( G )and s i r i ∈ E ( G ), and G has no other edges.We call a graph G a k - almost - skinny - ladder if the following conditions hold: • V ( G ) = L ∪ S ∪ R with L , S , and R mutually disjoint and | S | = k . • G [ L ] and G [ R ] form induced paths of G and L is anti-complete with R . • Each s ∈ S has at least one neighbor in L and at least one neighbor in R . • For all pairs x, y ∈ S , if a, b are neighbors of x in L , then y has no neighbors on the subpathof G [ L ] that has a and b as its endpoints. Similarly, if a, b are neighbors of x in R , then y hasno neighbors on the subpath of G [ R ] that has a and b as its endpoints. This last conditionrequires that no vertex of L or R has more than one neighbor in S .6he following graphs, except for k -ladder graphs, appear in Theorem 2. Figure 2 depicts thesegraphs. It can be seen that all graphs here except for k -claw and k -paw graphs contains at least2 k − minimal separators. • A graph G is a k - theta if G consist of two vertices a, b and k induced paths P , P , . . . P k . For1 ≤ i ≤ k the end points of P i are a and b , every P i is anti-complete with P j , and every P i has length at least 4. • A graph G is a k - prism if G consist of two disjoint cliques a , a , . . . , a k and b , b , . . . , b k along with k induced paths P , P , . . . P k each of length at least 2. For 1 ≤ i ≤ k the endpoints of P i are a i and b i , every P i − { a i , b i } is anti complete with P j − { a j , b j } , and a i isneighbors with b j if and only if i = j and P i is a path of length 2. • A graph G is a k - pyramid if G consist of a vertex a and a clique b , b , . . . , b k , where a isanti-complete with b , b , . . . , b k , along with k induced paths P , P , . . . P k each of length atleast 3. For 1 ≤ i ≤ k the end points of P i are a and b i and every P i − { a, b i } is anti completewith P j − { a, b j } . • A graph G is a k - ladder - theta if G consists of an induced path L and a vertex b anti-completewith L , along with k mutually disjoint induced paths P , P , . . . P k that are also disjoint from L , each of length at least 3. For 1 ≤ i ≤ k the end points of P i are a i and b , every P i − { b } is anti-complete with P j − { b } , P i − { a i } is anti-complete with L , every a i has at least oneneighbor in L , and if x, y are neighbors with a i in L , then no a j with i (cid:54) = j has a neighbor inthe induced subpath of L that has x and y as its endpoints. • A graph G is a k - ladder - prism if G consists of an induced path L and clique b , b , . . . , b k where L is anti-complete with b , b , . . . , b k , along with k induced paths P , P , . . . P k eachof length at least 2. For 1 ≤ i ≤ k the end points of P i are a i and b i , every P i − { b i } isanti-complete with P j − { b } , P i − { a i } is anti-complete with L , every a i has at least oneneighbor in L , and if x, y are neighbors with a i in L , then no a j with i (cid:54) = j has a neighbor inthe induced subpath of L that has x and y as its endpoints. • A graph G is a k - ladder if G consists of two disjoint paths L and R , along with k disjointinduced paths P , P , . . . P k each of length at least 2. For 1 ≤ i ≤ k the end points of P i are a i and b i , every P i − { b i } is anti-complete with P j − { b i } , P i − { a i } is anti-complete with P j − { a j } , every a i has at least one neighbor in L , every b i has at least one neighbor in R , if x, y are neighbors with a i in L , then no a j with i (cid:54) = j has a neighbor in the induced subpathof L that has x and y as its endpoints, and if x, y are neighbors with b i in R , then no b j with i (cid:54) = j has a neighbor in the induced subpath of R that has x and y as its endpoints. Notethat we do not have any requirements on the ordering that the neighborhoods the a i ’s and b i ’s have into L and R respectively (so it could happen that a i ’s neighborhood in L may lie inbetween a j ’s and a (cid:96) ’s neighborhood in L , but b i ’s neighborhood in R does not lie in between b j ’s and b (cid:96) ’s neighborhood in R , this is illustrated in the k -ladder given in Figure 2. We couldforce this not to happen though with an easy application of the Erdos-Szekers Lemma at thecost of making it a √ k -ladder.) • A graph G is a k - claw if G consists of k disjoint, anti-complete copies of the following graphwhich we call a long - claw of arm length k : let v be a vertex and P , P , P be three pathsof length k each with v as one of its endpoints and P i − { v } is anti-complete and disjoint with P j − { v } (i.e. the graph is a claw with each edge subdivide k − A graph G is a k - paw if G consists of k disjoint, anti-complete copies of the following graphwhich we call a long - paw of arm length k : let v , v , v be a triangle and P , P , P bethree disjoint induced paths of length k each such that P i has v i as one of its endpoints and P i − { v i } is anti-complete with P j − { v j } for 1 ≤ i (cid:54) = j ≤ k -Creature-Free Feral Graph Family Figure 3:
The k -twisted-ladder. In this section we will show that the graph of Figure 3,which we will refer to as the k -twisted ladder, is a coun-terexample to Conjecture 1. We begin next paragraph bygiving a few definitions, then in the following paragraphwe will observe that the k -twisted-ladder has 2 k minimalseparators, and finally Lemma 1 completes the counterex-ample by showing that the k -twisted-ladder does not con-tain a large k -creature.We defined a partition of the vertices as follows, let S denote the set of labeled vertices of the k -twisted ladderthat have 1 as their superscript. If we remove S from the k -twisted-ladder we get two induced path, one on the leftside which we will refer to as L and one on the right sidewhich we will refer to as R . We also define the i th block ofthe k -twisted ladder to be the set of vertices that containsthe vertices of the subpath of L that has c Li +1 and c Li asits endpoints, the vertices of the subpath of R that has c Ri +1 and c Ri as its endpoints, and the vertices a i and b i .So, the i th block and the ( i + 1) th block overlap at thevertices c Ri +1 and c Li +1 .To see that the k -twisted ladder has at least 2 k mini-mal separators we make a set X . For each i with 1 ≤ i ≤ k we choose j ∈ { , } and add a ji and b ji to X . X is then an x, y -minimal separator, and there are 2 k different choiceswe had when making X , so k -twisted ladder has at least2 k minimal separators.To complete the counterexample, we show in the fol-lowing lemma that this structure does not have a large k -creature. To make the result as easy as possible toverify, we show no k -twisted-ladder has a 100-creature,although a significantly smaller upper bound exists. Lemma 1. k -twisted-ladders are 100-creature-free for allk.Proof. Let H be a k -twisted ladder. Assume for acontradiction that H contains a 100-creature ( A, B , { x , x , . . . , x } , { y , y , . . . , y } ).Let X A and X B denote the highest numbered block that A and B have a vertex in respectively,and let Y A and Y B denote the lowest numbered block that A and B have a vertex in respectively.Let i = max ( Y A , Y B ) + 1 and let j = min ( X A , X B ) −
1. Let k be an integer such that i ≤ k ≤ j (If8o such k exists, then the only blocks that can contain vertices from both A and B must be twoadjacent blocks and it is clear the lemma is true in this case). Then since A and B are connected wecan see by inspection that A must contain one vertex from { c Lk , c Rk } and { c Lk +1 , c Rk +1 } and B mustcontain one vertex from { c Lk , c Rk } and { c Lk +1 , c Rk +1 } . Furthermore, since A is anti-complete with B ,we can again see from inspection that if c Lk ∈ A then we must have c Lk +1 ∈ A , c Rk ∈ B , and c Rk +1 ∈ B (since removal of the closed neighborhood of any path from c Lk to c Rk +1 , in fact even just the removalof the closed neighborhoods of c Lk and c Rk +1 , would separate blocks numbered greater than k fromblocks numbered less than k ), and if c Rk ∈ A then we must have c Lk ∈ B , and by the same reasoningit follows that c Lk +1 ∈ B , and therefore c Rk +1 ∈ A . Therefore, without loss of generality we mayassume that for all k with i ≤ k ≤ j that c Lk ∈ A and c Rk ∈ B . It then follows from this assumptionand the fact that A is anti-complete with B that there are only two possibilities for the restrictionof A and B to the k th block. Either we have that both the restriction of A is the subpath of L with endpoints c Lk and c Lk +1 and the restriction of B is the subpath of R with endpoints c Rk and c Rk +1 or both the restriction of A is the induced path made up of c Lk +1 along with b k and b k ’s twoneighbors in L and the restriction of B is the induced path made up of c Rk along with a k and a k ’stwo neighbors in R .By the definition of a k -creature, no vertex of the 100-creature can belong to N [ A ] ∩ N [ B ].Hence, by what was just shown, the restriction of the 100-creature to the blocks numbered betweenand including i and j induced two disjoint paths. Since the i − th and j + 1 th blocks are the onlytwo other blocks that can contain vertices from both A and B , it is now easy to see that ( A, B , { x , x , . . . , x } , { y , y , . . . , y } ) cannot be a 100-creature. k -Creature and k -Skinny-Ladder Induced Minor Free Graphs In this section we will provide all the lemmas needed for a proof of Theorem 1 and conclude thissection with said proof. We begin this section by stating some well known results. Corollary 1 thenshows that the neighborhood of a vertex v of a k -creature free graph G can intersect the minimalseparators of G that do not contain v in at most n k +1 different ways. Lemma 15 shows that theminimal separators of graphs that are k -creature-free and do not contain a k -skinny-ladder as aninduced minor can be dominated by a constant number vertices. Lemma 16 then uses a branchingalgorithm to list all minimal separators of its input graph assuming the input graph satisfies certainproperties and proves a bound on the number of minimal separators produced by this algorithm.This lemma is combined with Corollary 1 and Lemma 15 to give a proof of Theorem 1. Most ofthe work of this section goes into proving lemmas needed for the proof of Lemma 15. Lemma 2 ( Ramsey’s Theorem). [Ram30]For every pair of positive integer k and (cid:96) there is a least positive integer R ( k, (cid:96) ) such that everygraph with at least R ( k, (cid:96) ) vertices contains a clique of size k or an independent set of size (cid:96) . Throughout this paper we will us the notation R ( k, (cid:96) ) to denote the least positive integer suchthat every graph with at least R ( k, (cid:96) ) vertices contains a clique of size k or an independent set ofsize (cid:96) . Lemma 3 ( Erdos-Szekers Theorem). [ES09]For every pair on positive integers r and s , any sequence of distinct real numbers of lengthat least ( r -1)( s -1) + 1 contains a monotone increasing subsequence of length r or a monotonedecreasing subsequence of length s . efinition 1 ( V.C. Dimension).
Let F = { S , S , . . . } be a family of sets and let H be a set. F is said to shatter H if for every subset H (cid:48) ⊆ H there is a S i ∈ F such that H (cid:48) = S i ∩ H . The V.C. - dimension of F is the cardinality of the largest set that it shatters. Lemma 4 ( Sauer-Shelah Lemma). [Sau72]Let F be a family of sets such that the V.C.-dimension of F is k , and let n = | (cid:83) S i ∈F S i | , so n is the number of distinct elements contained in the sets of F . Then the number of sets of F is atmost Σ ki =0 (cid:0) ni (cid:1) ≤ n k + 1. Given a graph G and two non adjacent vertices u, v ∈ G , we say that a u, v -minimal separator S is close to v if S ⊆ N ( v ). The following two lemmas establish useful properties of u, v -minimalseparators close to v . Lemma 5. [KK94] Given a graph G and two non adjacent vertices u, v ∈ V ( G ) , there exists aunique u, v -minimal separator that is close to v . Let S and S (cid:48) be two u, v minimal separators for a graph G . Let C u be the connected component u belongs to in G − S and let C (cid:48) u be the connected component u belongs to in G − S (cid:48) . Then wesay that S (cid:48) ≤ u,v S if C (cid:48) u ⊂ C u . Lemma 6.
Let S v,u be the u, v -minimal separator close to v given by Lemma 5 for some graph G and let S be another u, v -minimal separator. Then S ≤ u,v S v,u .Proof. Let S v,u be the u, v -minimal separator close to v given by Lemma 5 for some graph G andlet S be another u, v -minimal separator. Assume for a contradiction that S ≤ u,v S v,u does nothold. Let C u be the connected component u belongs to in G − S v,u and let C (cid:48) u be the connectedcomponent u belongs to in G − S . Then there is some vertex x ∈ C (cid:48) u that is not in C u . Since x isnot in C u it follows that every path from u to x must contain a vertex from S v,u , but this impliesthat there must then be some vertex from S v,u that belongs to C (cid:48) u since x ∈ C (cid:48) u which means that S would not separate v from u (since every vertex in S v,u is neighbors with v ), a contradiction. Lemma 7. If S is a u, v -minimal separator for a graph G that is k -creature free and S + = { S (cid:48) ∩ S : S (cid:48) ≤ u,v S, S (cid:48) is a u, v -minimal separator } , then |S + | ≤ | V ( G ) | k .Proof. Let S be a u, v -minimal separator of a k -creature free graph G , let H be the S -full componentof G − S that contains u , and let | S + | be as in the statement of the lemma. We show thatthe V.C. dimension of | S + | is less than k . It will then follow by the Sauer-Shelah Lemma that | S + | ≤ | V ( G ) | k − +1 ≤ | V ( G ) | k (assuming | V ( G ) | >
1, but the lemma is trivially true if | V ( G ) | = 1or 0).Assume for a contradiction that the V.C. dimension of | S + | is at least k . Then there existsminimal separators S , S , . . . , S k and vertices s , s , . . . , s k ∈ S such that S ∩ S i = { s i } and S i ≤ u,v S . Let H i denoted the S i -full component u belongs to in G − S i , let s (cid:48) i be a neighbor of s i in H i such that its distance from u in H i is minimum among all neighbors of s i in H i , and let P i denote a shortest path from s (cid:48) i to u that is contained completely in H i . It follows that the onlyvertex of P i that is neighbors with s i is s (cid:48) i . Let P = (cid:83) ≤ i ≤ k P i , and let K be the S -full componentthat contains v . We then have that ( V ( K ), V ( P ) − { s (cid:48) , s (cid:48) , . . . , s (cid:48) k } , { s , s , . . . , s k } , { s (cid:48) , s (cid:48) . . . , s (cid:48) k } )forms a k -creature. To see this note that P and K are anti-complete since S i ≤ u,v S , so all thatmust be verified is that if s i s (cid:48) j ∈ E ( G ) then i = j and that P − { s (cid:48) , s (cid:48) , . . . , s (cid:48) k } and { s , s , . . . , s k } are anti-complete. We already saw before that s (cid:48) i is the only vertex of P i that is neighbors with s i . Furthermore, if there is some element p j ∈ P j such that s i p j ∈ E ( G ) but i (cid:54) = j then thisimplies that s i ∈ H j since s i / ∈ S j , but this would contradict the fact that S j ≤ u,v S . It follows P − { s (cid:48) , s (cid:48) , . . . , s (cid:48) k } is disjoint from { s , s , . . . , s k } and that s i s (cid:48) j ∈ E ( G ) if and only if i = j .10s noted before, the following corollary is a key part of the proof of Theorem 1. Corollary 1. If G is a k -creature-free graph, then for every v ∈ V ( G ) , with S v = { N ( v ) ∩ S : v / ∈ S and S is a minimal separator of G } , it holds that | S v | ≤ | V ( G ) | k +1 .Proof. Let G be a k -creature-free graph, and for every v ∈ V ( G ) let S v be as in the statement ofthis lemma. For each u in G with u (cid:54) = v and uv / ∈ E ( G ), let S v,u be the u, v -minimal separatorclose to v given by Lemma 5, and let S vu = { S v,u ∩ S : S is a u, v -minimal separator } . Since forevery u, v -minimal separator, S (cid:48) , it holds that S (cid:48) ≤ u,v S v,u by Lemma 6, applying Lemma 7 using S v,u as S , it follows that S vu ≤ | V ( G ) | k . We will show that S v = (cid:83) S vu where the union is takenover all u ∈ V ( G ) with u (cid:54) = v and uv / ∈ E ( G ). It will then follow that | S v | ≤ | V ( G ) | k +1 .Let S be some minimal separator such that v / ∈ S , let C be the component v is in in G − S ,and let u be a vertex in some S -full component of G − S different from C , so S is a u, v -minimalseparator. Now, if w ∈ N ( v ) and w ∈ S , then since u ’s component of G − S is S full there is apath from w to u that contains no vertex of C or S other than w and therefore contains no vertexof N ( v ) other than w . So if S v,u did not contain w then it would not separate u from v , hence w ∈ S v,u . Since S v,u ⊂ N ( v ) it follows that S v,u ∩ S = N ( v ) ∩ S . Hence, N ( v ) ∩ S ∈ S vu , and theresult follows.The following corollary will be needed in Sections 6 and 7 Corollary 2. If G is a k -creature-free graph and every minimal separator, S , of G can be dominatedby k vertices of G not in S , then G has at most | V ( G ) | k +2 k minimal separators.Proof. Assume G is a k -creature-free graph and every minimal separator, S , of G can be dominatedby k vertices of G not in S . For every v ∈ V ( G ) let S v = { N ( v ) ∩ S : v / ∈ S and S is a minimalseparator of G } . By Corollary 1 it holds that | S v | ≤ | V ( G ) | k +1 . Let X = (cid:83) v ∈ G S v . Then | X | = | V ( G ) | k +2 and the assumption that all minimal separators, S , of G can be dominated by k vertices in G not in S implies that S is the union of at most k sets in X . It follows there are atmost | V ( G ) | k +2 k minimal separators in G .We remark that it is possible to generalize Corollaries 1 and 2 to the r th neighborhood of avertex for any fixed positive integer r while still maintaining polynomial bounds by using the factthe family of k -creature-free graphs are closed under contracting edges.The following lemmas will be building towards a proof of Lemma 15. We begin with an easyobservation that will be useful in the proof of Lemma 15. Lemma 8.
Let G be a directed graph with maximum out-degree or maximum in-degree at most c , c > . Then G has an independent set (no vertex is an in-neighbor or out-neighbor of any othervertex in this set) of size at least | V ( G ) | c +1 . Furthermore, if | V ( G ) | ≥ t and the maximum out-degreeor maximum in-degree of G is at most t | V ( G ) | , then G has an independent set of size at least t .Proof. Let G be a directed graph. We will prove the statements for bounded maximum out-degree(for maximum in-degree the proof is nearly identical). If the maximum out-degree of G is c , c > G has at least one vertex, there must exists a vertex v ∈ G with in-degree at most c . If we let G (cid:48) be the subgraph induced by all vertices of G − v that do not have v as an in-neighboror an out-neighbor, then the size of G (cid:48) is at least | V ( G ) | - 2 c −
1, and G (cid:48) has maximum degree c .It follows by an inductive argument that we can find an independent set of size at least | V ( G ) | c +1 .To prove the furthermore statement, assume the maximum out degree of G is at most t | V ( G ) | , t >
0, and | V ( G ) | ≥ t , so we have that | V ( G ) | t + 1 ≤ | V ( G ) | t . From the first paragraph we have that G contains an independent set of size at least | V ( G ) | | V ( G ) | t +1 = | V ( G ) | | V ( G ) | t +1 ≥ | V ( G ) | | V ( G ) | t = t .11emmas 9, 10, and 11 are used to help in the proof of Lemma 12. Lemma 12 then is usedto produce a structure that is similar to a k -skinny-ladder in graphs that are k -creature free andhave minimal separators that cannot be dominated by few vertices. The structure that Lemma 12produces is then used in Lemma 15 to produce a k -skinny-ladder as an induced minor in graphsthat are k -creature-free and have minimal separators that cannot be dominated by few vertices. Lemma 9.
Let G be a graph that is k -creature free, let S be a minimal separator of G , and let A be an S -full component of G − S . Then S is dominated by the union of less than k induced pathsin A .Proof. Let G , S , and A be as in the statement of this lemma, and let A (cid:48) be a minimally connectedinduced subgraph of A such that S is dominated by A (cid:48) . Let T be a breadth first search tree of A (cid:48) rooted at some vertex v ∈ A (cid:48) , and let L = { (cid:96) , (cid:96) , . . . , (cid:96) c } be the set of leaves of T . Since A (cid:48) isminimal each leaf, (cid:96) i ∈ L , must have a neighbor s i ∈ S such that no other vertex of A (cid:48) is neighborswith s i , else A (cid:48) − s i would still be connected and dominate S . Then if K is another S -full componentdifferent from A the tuple ( V ( A (cid:48) ) − L, V ( K ) , { (cid:96) , (cid:96) , . . . , (cid:96) c } , { s , s , . . . , s c } ) forms a c -creature. Itfollows that if G is k -creature-free, then T has at most k − A (cid:48) is theunion of at most k − S, H, P of a k -creature-free graph G and a vertex v ∈ G and finds a small set X such that no vertex of S − N [ X ]shares a common neighbor with v in P or is neighbors with v . Note that in Lemma 10 v may ormay not be in S . Lemma 10.
Let G be a graph and ( G, S, H, P, v ) be a tuple with the following properties: S ⊂ V ( G ) , v ∈ G , H and P are induced subgraphs of G such that H is connected, P is an induced path, S ∪{ v } , V ( H ) , V ( P ) are mutually disjoint, H is anti-complete with P , v has no neighbor in H , and allvertices in S − { v } have a neighbor in H . Then if G does not contain a k -creature, there is a set, X ⊂ S ∪ V ( H ) ∪ V ( P ) , of size at most k − such that N ( S − N [ X ]) ∩ N ( v ) ∩ V ( P ) = ∅ , and novertex of S − N [ X ] is neighbors with v .Proof. Let G , S , H , P , and v be as in the statement of this lemma. Number the vertices of P | V ( P ) | so that the vertex numbered i is neighbors with the vertices numbers i − i + 1.We now consider the following process to build the set desired set X such that N ( S − N [ X ]) ∩ N ( v ) ∩ V ( P ) = ∅ .We do the following for the first step of the process. Let X = { v } , and let S = { s : s ∈ S − N [ X ] and N ( s ) ∩ N ( v ) ∩ V ( P ) (cid:54) = ∅} (i.e. S is the set of vertices of S − N [ v ] that sharea neighbor with v in P ). Label the vertices of S by the lowest numbered vertex it is neighborswith in V ( P ) ∩ N ( v ). Let s be a highest labeled vertex in S , and let p be s ’s lowest numberedneighbor in N ( v ) ∩ V ( P ). This completes the first step.For the i th step, i >
1, we do the following. Let X i = X i − ∪ { s i − , p i − } , and let S i = S − N [ X i ]and label the vertices of S i by the lowest vertex it sees in V ( P ) ∩ N ( v ) (the vertices of S i inherittheir labels from S i − ). Let s i be a highest labeled vertex in S i and let p i be s i ’s lowest neighborin N ( v ) ∩ V ( P ). This completes the i th step. Note by how we selected v , s , p , s , p , . . . s i , p i that for 1 ≤ a, b ≤ i , s a cannot be neighbors with p b if a > b since p b would be in X a and therefore s a would not be in S a , and s a cannot be neighbors with p b if a < b since that would contradicteither p a being s a ’s lowest numbered neighbor in N ( v ) ∩ P or s a being a highest labeled vertex in S a . Hence, we then have that among these vertices s j is only neighbors with p j for 1 ≤ j ≤ i , and v is only neighbors with p j for 1 ≤ j ≤ i . 12e continue this process until we reach an S j that is empty. We claim this process cannot gopast the k th step if G does not contain a k -creature. Assume for a contradiction, that this processcompletes the k th step. We claim the tuple ( { v } , V ( H ) , { p , p , . . . , p k } , { s , s , . . . , s k } ), forms a k -creature. As noted before, by how we selected v , s , p , s , p , . . . s k , p k we have that amongthese vertices s j is neighbors with p j and not with p r for r (cid:54) = j , and v is only neighbors with the p j ’s. We also have that by assumption v has no neighbors in H , but all of the vertices s , s , . . . , s k have neighbors in H . Lastly, we can see that no vertex of p , p , . . . , p k has a neighbor in H by theassumption that P is anti-complete with H . It follows that this tuple is a k -creature.Set X to be X j , where j is the first iteration where S j is empty. Since S j is empty, it follows N ( S − N [ X ]) ∩ N ( v ) ∩ V ( P ) = ∅ . We also have that no vertex of S − N [ X ] is neighbors with v since v ∈ X and | X | ≤ k − j ≤ k and since the first step adds a single vertex and eachiteration after that only adds two vertices.Note that in Lemma 11, v may or may not be in S . Lemma 11.
Let G be a k -creature free graph and let ( G, S, H, P, v ) be a tuple with the followproperties: S ⊂ V ( G ) , v ∈ G , H and P are connected induced subgraphs of G , S ∪ { v } , V ( H ) , and V ( P ) are vertex disjoint, H is anti-complete with P , v has no neighbor in H or S , all vertices in S −{ v } have a neighbor in H and in P , and for all x ∈ S −{ v } it holds that N ( x ) ∩ N ( v ) ∩ V ( P ) = ∅ .Then there is a set of at most k − connected components of P − N ( v ) such that every vertex of S − { v } has a neighbor in at least one of these connected components.Proof. Let
G, S, H, P , and v be as in the statement of this lemma. Assume for a contradiction thatthere does not exists a set of at most k − P − N ( v ) such that everyvertex of S − { v } has a neighbor in at least one of these connected components. It follows thenthere is a set of k connected components of P − N ( v ), say C , C , . . . , C k , such that there exists s , s , . . . , s k in S where N ( s i ) ∩ V ( C j ) (cid:54) = ∅ if and only if i = j . Since P is connected, for every C i there exists a vertex c i ∈ N ( v ) ∩ V ( P ) such that c i ∈ N ( C i ) (the c i ’s may not be unique). Now,for each s i , let s (cid:48) i be a vertex in C i that s i is neighbors with such that there exists an induced path P i , with internal vertices in C i , from s (cid:48) i to c i such that s (cid:48) i is the only neighbor of s i on the path P i .Then the tuple ( { v } (cid:83) V ( P i − s (cid:48) i ) , H, { s (cid:48) , s (cid:48) . . . , s (cid:48) k } , { s , s . . . , s k } ) is a k -creature, contradictingthe assumption G is k -creature-free.The following lemma produces a structure similar to a k -skinny-ladder in graphs that are k -creature-free and contain minimal separators that cannot be dominated by few vertices. Thisstructure will be the main object of study in Lemma 15. Lemma 12.
Let S be a minimal separator of a graph G such that S cannot be dominated by k vertices. Then if G is k -creature there exists there exists a subset S (cid:48) of S of size k such that thereexists two paths, P and P , in two different components of G − S that dominate the vertices of S (cid:48) ,and no vertex of P or P has more than one neighbor in S (cid:48) .Proof. Assume that G is a k -creature free graph, and let S (cid:48) be a minimal separator of G thatcannot be dominated by 4 k vertices of G , and let L (cid:48) and R (cid:48) be two different S (cid:48) -full componentsof G . It follows from Lemma 9 that there is a set of less than k induced paths in L (cid:48) that togetherdominated S (cid:48) and there is a set of less than k induced paths in R (cid:48) that together dominate S (cid:48) . Itfollows there exists two induced paths L ⊂ L (cid:48) and R ⊂ R (cid:48) such that ( N ( L ) ∩ S (cid:48)(cid:48) ) ∩ ( N ( R )) ∩ S (cid:48) )cannot be dominated by 4 k vertices of G . Let S = ( N ( L ) ∩ S (cid:48) ) ∩ ( N ( R ) ∩ S (cid:48) ). Fix a numberingthe vertices of L | V ( L ) | so that the vertex numbered i is neighbors with the verticesnumbers i − i + 1. 13ssume that we have an independent set of vertices S i − of size i − i ≤ k , and a vertex set Z i − of size at most 4 k ( i − S − N [ Z i − ] is neighbors with avertex in S i − , and any vertex in L or R that is neighbors with some vertex in S i − has no otherneighbors in S i − or in S − N [ Z i − ]. We will show how to produce a set S i of size i and Z i of sizeat most 4 k i with the corresponding properties, assuming i ≤ k . Note that the empty set satisfiesthe condition of S .Let S (cid:48) = S − N [ Z i − ], and label the vertices of S (cid:48) according to the lowest numbered neighbor ithas in L . Let s be a highest labeled vertex in S (cid:48) , since i ≤ k and since S cannot be dominated by4 k vertices such an s must exists. Let (cid:96) be the lowest numbered neighbor s has in L and assumethe number of (cid:96) is p , and let H denote the subpath of L that is made up of the vertices labeled 1through p −
1. We can then apply Lemma 10 using (
G, S (cid:48) − N ( (cid:96) ) , H, R, s ) to get a set X (cid:48) of size atmost 2 k − S (cid:48) − N [ X ∪ { (cid:96) } ]) ∩ N ( s ) ∩ V ( R ) = ∅ and no vertex of S (cid:48) − N [ X ] is neighborswith s . Set X = X (cid:48) ∪ { (cid:96) } .We now wish to find a set Y of size less than 2 k so that no vertex of S (cid:48) − ( N [ X ] ∪ N [ Y ])shares a neighbor with s in either L or R . We first use Lemma 11 on ( G, S (cid:48) − N [ X ] , H, R, s ) to getconnected components C , C , . . . , C c , c < k , of R − N ( s ) such that all vertices of S (cid:48) − N [ X ] have aneighbor in at least one C i . Then for each C i we apply Lemma 10 on ( G , ( S (cid:48) − N [ X ]) ∩ N ( C i ), C i , L , s ) to get a set Y i of size less than 2 k such that no vertex of [( S (cid:48) − N [ X ]) ∩ V ( C i )] − N [ Y i ] shares aneighbor with s in L (or R ). It follows that if we set Y = (cid:83) Y i that no vertex of ( S (cid:48) − N [ X ]) − N [ Y ]shares a neighbor with s in L (or R ). We may then set S i = S i − ∪ { s } and Z i = Z i − ∪ X ∪ Y .Since | X | ≤ k and | Y | ≤ k we have that | Z i | ≤ | Z i − | + 2 k + 2 k ≤ k i .The statement of the lemma now follows from the fact that S cannot be dominated by 4 k vertices so this process may go on until we attain the set S k , which is the desired set, along withthe paths L and R .The next two Lemmas will be useful in the proof of Lemma 15. Lemma 13.
Let G be a graph that contains a k -almost-skinny-ladder as an induced subgraph.Then G contains a k -skinny-ladder as an induced minor.Proof. Let G be a graph that has a k -almost-skinny-ladder, H , as an induced subgraph. V ( H ) = L ∪ S ∪ R where L, S, R each have the same meaning as in the definition of an almost-skinny-laddergiven in Section 2. Number the vertices of L | V ( L ) | so that the vertex numbered i isneighbors with the vertices numbered i − i + 1, and similarly, number the vertices of R | V ( R ) | so that the vertex numbered i is neighbors with the vertices numbered i − i + 1.Next we label each vertex in S with a subscript 1 through | S | so that for all s i , s j ∈ S i > j if and only if all of s i ’s neighbors in L have a higher number than all of s j ’s neighbors in L (bythe definition of an almost-skinny-ladder such a numbering exists). Let n ( s i ) be the number ofthe highest numbered neighbor s i has is R . We now apply the Erodos-Szekers Theorem to thesequence n ( s ) , n ( s ) . . . , n ( s k ) to get an increasing or decreasing subsequence of length at least k and set S ∗ to be the subset of S that corresponds to the subsequence obtained from the Erodos-Szekers Theorem. If the Erodos-Szekers Theorem returned a decreasing subsequence then reversethe numbering of R , else leave it unchanged. Then for every s i , s j ∈ S ∗ , if i > j then all of s i ’sneighbors in L have a higher number than all of s j ’s neighbors in L and all of s i ’s neighbors in R have a higher number than all of s j ’s neighbors in R . We can now apply the obvious edgecontractions to L and R to form a k -skinny-ladder.14 emma 14. Let G be a graph, let a, b ∈ G be two non adjacent vertices of G , and let P , P , . . . , P k be K paths that are anti-complete with respect to one another and for every P i , both a and b havea neighbor in P i . Furthermore assume that for every P i that no vertex of P i is neighbors with both a and b . Then G contains a k -theta.Proof. Let G , a, b , P , P , . . . , P k be as in the statement of the lemma. For each P i we can then, byassumption, find a subpath of P i , call it P ∗ i , such that P ∗ i has endpoints a i , b i where a i is neighborswith a , b i is neighbors with b , no internal vertex is neighbors with a or b . It follow again byassumption that each P ∗ i has length at least 2 and that together the P ∗ i ’s along with a and b makea k -theta.The following lemma essentially takes a k -creature-free graph G that has a minimal separatorthat cannot be dominated by few vertices, obtains the structure given by Lemma 12 and cleans itup to produce k -skinny-ladder as an induced minor of G . Lemma 15.
Let S be a minimal separator of a graph G such that S cannot be dominated by k ) k +1 ] vertices. If G is k -creature-free, then G contains a k -ladder as an induced minor.Proof. Assume that G is k -creature-free and S (cid:48) is a minimal separator of G such that S (cid:48) cannot bedominated by 4[(8 k ) k +1 ] vertices. It follows from Lemma 12 that there is a set S ⊂ S (cid:48) of (8 k ) k +1 vertices and two paths R and L that dominate S , L anti-complete with R , and every vertex in v ∈ V ( L ) ∪ V ( R ) has at most one neighbor in S .Number the vertices of L | V ( L ) | so that the vertex numbered i is neighbors with thevertices numbered i − i + 1, and number the vertices of R | V ( R ) | so that the vertexnumbered i is neighbors with the vertices numbered i − i + 1. For a vertex x in L or R wewill use the notation n ( x ) to denote the number it has been given in L or R . For every s j ∈ S let (cid:96) j ∈ L and r j ∈ R be the highest numbered neighbors of s j in L and R respectively. We nowset L = L, R = R , and S = S . We will consider the following process to produce a k -almost-skinny-ladder. We will show this process cannot go past k iterations if G is k -creature-free, and wewill ensure that at the i th step that V ( L i ) ⊂ V ( L ), V ( R i ) ⊂ V ( R ), S i ⊂ S , | S i | ≥ (8 k ) k − i +2 , L i and R i are induced paths, and if s j ∈ S i then (cid:96) j ∈ L i and r j ∈ R i . We will also produce inducedsubpaths P i of either L or R such that the P i ’s are anti-complete with respect to one another andthe vertices of P i will dominate S j if i < j .At the i th step, i ≤ k , we do as follows. Create an auxiliary directed graph, AU X i , whose vertexset is S i and there is an edge from s a ∈ S i to s b ∈ S i if at least one of the following two cases hold1. n ( (cid:96) a ) > n ( (cid:96) b ) and s a has a neighbor x in L such that n ( x ) < n ( (cid:96) b )2. n ( r a ) > n ( r b ) and s a has a neighbor x in R such that n ( x ) < n ( r b )If the maximum in-degree of AU X i is at most k | S i | then we stop. Since | S i | ≥ (8 k ) k − i +2 this gives an independent set of size at least k by Lemma 8. If there is an s j ∈ S i with in degreeover k | S i | then for at least k fraction of the vertices of S i , call this subset of vertices S i +1 , allvertices s ∈ S i +1 must satisfy case 1 one with s playing the role of s a and s j playing the role of s b , or all vertices s ∈ S i +1 must satisfy case 2 again with s playing the role of S a and s j playingthe role of s b . For each case we now describe what to do if all the vertices of S i +1 satisfy thatcase (if all vertices of S i +1 satisfy both cases, then we go with the first case). Each number herecorresponds what to do in that case.1. Call P i the subpath of L i that is made up of vertices with numbers less than n ( (cid:96) j ). Set R i +1 = R i and set L i +1 to be the vertices of L with numbers greater than n ( (cid:96) j ).15. Call P i the subpath of R i that is made up of vertices with numbers less than n ( r j ). Set L i +1 = L i and set R i +1 to be the vertices of R with numbers greater than n ( r j ).It can then be seen that V ( L i +1 ) ⊂ V ( L ), V ( R i +1 ) ⊂ V ( R ), S i +1 ⊂ S , | S i +1 | ≥ (8 k ) k − i +1 , L i +1 and R i +1 are induced paths, and if s j ∈ S i +1 then (cid:96) j ∈ L i +1 and r j ∈ R i +1 as required.Furthermore, it can be seen that any of the previously P j ’s that have been produced in this process( j ≤ i ) dominate all vertices of S i +1 and are anti-complete with respect to one another. By Lemma14 then, this process cannot go past the k th iteration without producing a k -theta.We conclude there is some step j ≤ k such that the auxiliary graph AU X j has max in-degreeless than k | S j | , and since | S j | ≥ k it therefore has an independent set of size k by Lemma 8.Let S ∗ denote such an independent set, we claim that G [ V ( L ) ∪ S ∗ ∪ V ( R )] makes an k -almost-skinny-ladder. Let x, y ∈ S ∗ and let a, b be the highest and lowest numbered neighbors of x in L respectively, and assume that y has a neighbor c on the induced path of L that has a and b as its endpoints. If y ’s highest numbered neighbor in L is greater than n ( a ) then y has an edgeto x in AU X j by case (1). If y ’s highest numbered neighbor is L is less than n ( a ), then x hasan edge to y again by case (1). Both cases yield a contradiction to S ∗ being an independent setin AU X j . A nearly identical argument show that if a (cid:48) , b (cid:48) are x ’s highest and lowest numberedneighbors R respectively, then y cannot have a neighbor in the induced subpath of R that has a (cid:48) , b (cid:48) as its endpoints. It follows that G [ V ( L ) ∪ S ∗ ∪ V ( R )] is a k -almost-skinny-ladder. ApplyingLemma 13 shows that G contains a k -skinny-ladder as an induced minor.The following lemma uses a branching algorithm to produce all of the minimal separators ofa graph G and proves a bound on the number of minimal separators produced by this algorithm,which when combined with Corollary 1 and Lemma 15 gives a proof of Theorem 1. Lemma 16.
There exists a function f : N → N such that the following holds. Let G be a graphand let k and c be integers such that for all induced subgraphs G (cid:48) of G and for all v ∈ V ( G (cid:48) ) ,if S vG (cid:48) = { N ( v ) ∩ S : v / ∈ S and S is a minimal separator of G (cid:48) } , then | S vG (cid:48) | ≤ c and everyminimal separator of any induced subgraph of G can be dominated by k vertices. Then G has amost ( c + n k ) f ( k ) log( n ) minimal separators where n = | V ( G ) | .Proof. Let G , S vG , k , and c be as in the statement of this lemma. The proof of the bound makesuse of a branching algorithm. The algorithm takes as input G and X ⊂ V ( G ) and the algorithmwill use the set K ret to store the vertex sets it will return. It will return K ret which will containall minimal separators of G contained in X (most likely along with other vertex sets). We have noconcern about the runtime of the algorithm, but we care about the size of the final set it returns.The algorithm is intended to be used initially on the input ( G , V ( G )).Assume the the input to the algorithm is ( G, X ). If X is empty, then the algorithm returns ∅ .Else, the algorithm determines the set Q ⊂ V ( G ) where Q contains all vertices v ∈ V ( G ) such that | N [ v ] ∩ X | ≥ k | X | . Then the algorithm branches in the following two ways:1. For every q ∈ Q and every Y ∈ S qG the algorithm recursively calls itself on ( G − Y , X − N G [ q ]).Each recursive call returns a set K (cid:48) , which contains vertex sets. Then if the recursive call( G − Y , X − N G [ q ]) returns the collection K (cid:48) of vertex sets, for each set S in K (cid:48) , the algorithmadds the set S ∪ Y to K ret .2. For every set R of k vertices of G such that R ∩ Q = ∅ , the algorithm recursively calls itselfon ( G − Q , ( X ∩ N G ( R )) − Q ). Each recursive call returns a set K (cid:48) , which intern containsvertex sets. Then for each set, S , in each K (cid:48) returned the algorithm adds the set S ∪ Q to K ret . 16fter completing this, the algorithm then returns the set K ret . Note that in (2) since the set R has no vertex in Q and | R | ≤ k , the neighborhood of R contains at most of the vertices of X .Since Q contains all vertices v ∈ V ( G ) such that | N [ v ] ∩ X | ≥ k | X | , each recursive call thealgorithm makes is on input ( G (cid:48) , X (cid:48) ) where | X | ≥ (1 − k ) | X (cid:48) | , so the algorithm terminates. Let S be a minimal separator of G contained in X . Assume all of the recursive calls ( G (cid:48) , X (cid:48) ) the algorithmmakes returns a set that contains all minimal separators of G (cid:48) contained in X (cid:48) . If Y = N G ( q ) ∩ S forsome q ∈ G and q / ∈ S , then S − Y is a minimal separator of G − Y that is contained in X − N G [ q ].So if there is a q ∈ Q such that q / ∈ S , then S gets added to K ret in (1). If Q ⊂ S , then S − Q is aminimal separator of G − Q , and by assumption there exists some collection of at most k vertices, R , in G − Q such that S − Q ⊂ N G − Q ( R ) and therefore S − Q ⊂ ( X ∩ N G ( R )) − Q . It follows thatin this case we also have S gets added to K ret in (2). Induction on the the depth of the recursivecall now shows that this algorithm returns all minimal separators.If T ( n, x ) represents the maximum number of minimal separators that a vertex set X of size atmost x can contains for any graph G with | V ( G ) | ≤ n and X ⊂ V ( G ), such that the graph G satisfiesthe conditions of the lemma, then the algorithm shows that T ( n, x ) ≤ ( c + n k ) T ( n, [1 − k ] x ). Usingthe fact that lim y →∞ (1 − y ) y = e we expand the inequality T ( n, x ) ≤ ( c + n k ) T ( n, [1 − k ] x ) out O ( k )times to get T ( n, x ) ≤ ( c + n k ) O ( k ) T ( n, x ). Since T ( n,
0) = 0 it follows that there exists a function f : N → N (independent of the choice of k or G ) such that this solves to T ( n, x ) ≤ ( c + n k ) f ( k ) log( x ) .By taking the initial X to be V ( G ), it follows that G then contains at most ( c + n k ) f ( k ) log( n ) minimal separator, where n = | V ( G ) | .We are now ready to prove Theorem 1. Proof of Theorem 1.
Let G be a graph, | V ( G ) | = n , that is k -creature-free and has no k -skinny-ladder as an induced minor. For every induced subgraph G (cid:48) of G and for every v ∈ G (cid:48) , let S vG (cid:48) = { N ( v ) ∩ S : v / ∈ S and S is a minimal separator of G (cid:48) } . Then | S vG (cid:48) | = n f ( k ) for some function f : N → N by Corollary 1 ( f is independent of the choice of k or G ). By Lemma 15, since G is k -creature free and has no k -skinny-ladder as an induced minor, there is a function f (cid:48) : N → N ( f (cid:48) is independent of the choice of k or G ) such that every minimal separator of any induced subgraphof G (cid:48) is dominated by f (cid:48) ( k ) vertices. Lemma 16 then implies there is a function f (cid:48)(cid:48) : N → N ( f (cid:48)(cid:48) is independent of the choice of k or G ) such that G has at most ( f ( k ) + n f (cid:48) ( k ) ) f (cid:48)(cid:48) ( k ) log( n ) minimalseparators. We can then see there exists a function f ∗ : N → N ( f ∗ is independent of the choiceof k or G ) such that G has at most n f ∗ ( k ) log( n ) minimal separators. It follows that the familyof graphs that are k -creature-free and do not contain a k -skinny-ladder as an induced minor arestrongly-quasi-tame. In this section we will provide the lemmas needed in the proof of Theorem 2 as well give a proofof Theorem 2 at the end of this section. The majority of the work of this section goes into provingthat given an integer k , if G contains a k (cid:48) -creature for large enough k (cid:48) , then G must contain a k -theta, k -prism, k -pyramid, k -ladder-theta, k -ladder-prism, or k -ladder as an induced subgraph,which is proven in Lemma 25. Lemmas 28 and 29 then show that if F is a family of graphs definedby a finite number of forbidden induced subgraphs and F does not forbid all k -theta, k -prism, k -pyramid, k -ladder-theta, k -ladder-prism, k -claw, k -paw graphs for all k larger than some fixedconstant, then F is feral. Theorem 2 is then proved using Lemma 25 along with Theorem 1 and afew simple observations, as well as Lemmas 28 and 29.17t will be useful in this section to define the following graphs. These graphs are depicted inFigure 4. • A graph G is a k - half - theta if G consists of a vertex v and k induced paths P , P , . . . , P k of G such that each path has length at least 2, for 1 ≤ i ≤ k it holds that v is one endpoint of P i ,and for j (cid:54) = i it hold that P i − v is anti-complete with P j − v . Let x i denote the endpoint of P i that is not v . Then we say the vertices x , x , . . . , x k are the endpoints of the k -half-theta.If X is a vertex set and x i ∈ X for all i with 1 ≤ i ≤ k , then we say G is a k -half-theta endingin X . • A graph G is a k - half - prism if G consists of a clique of vertices v , v , . . . , v k and k inducedpaths P , P ,..., P k of G such that each path has length at least 1, for 1 ≤ i ≤ k it holds that v i is one endpoint of P i , and for j (cid:54) = i it hold that P i − v i is anti-complete with P j . If thelength of P i is greater than 1 then let x i denote the endpoint of P i that is not v i , and if thelength of P i is 1 then let x i = v i . We say the vertices x , x , . . . , x k are the endpoints of the k -half-prism. If X is a vertex set and x i ∈ X for all i with 1 ≤ i ≤ k , then we say G is a k -half-prism ending in X . • A graph G is a k - half - ladder if G consists of a path P of G along with k additional paths P , P , . . . , P k of G such that each path has length at least 1. For 1 ≤ i ≤ k let P i ’s endpointsbe v i and x i (with v i possibly equal to x i ). We call P the backbone path and the P i ’s theauxiliary paths. We require that v i has at least one neighbor in P , P is anti-complete with P i − v i , and for j (cid:54) = i P i is anti-complete with P j . Lastly, we also require that if a and b are two neighbors of some v i in P , then there is no v j , i (cid:54) = j such that v j has a neighbor inthe induced subpath of P with endpoint a and b . We say the vertices x , x , . . . , x k are theendpoints of the k -half-ladder. If X is a vertex set and x i ∈ X for all i with 1 ≤ i ≤ k , thenwe say G is a k -half-ladder ending in X . • A graph G is a k - half - quasi - ladder if G consists of a path P of G along with k additionalpaths P , P , . . . , P k of G such that each path has length at least 1. For 1 ≤ i ≤ k let P i ’sendpoints be v i and x i (with v i possibly equal to x i ). We call P the backbone path and the P i ’s the auxiliary paths. We require that v i has at least one neighbor in P , P is anti-completewith P i − v i , and for j (cid:54) = i P i is anti-complete with P j . We say the vertices x , x , . . . , x k are the endpoints of the k -half-quasi-ladder. If X is a vertex set and x i ∈ X for all i with1 ≤ i ≤ k , then we say G is a k -half-ladder ending in X . Note that a k -half-quasi-ladderis almost the same as a k -half-ladder, but we drop the requirement that if a and b are twoneighbors of some v i in P , then there is no v j , i (cid:54) = j such that v j has a neighbor in thesubpath of P with endpoint a and b .The following lemmas, culminating with Lemma 25, work towards proving that given an integer k , if G contains a k (cid:48) -creature for large enough k (cid:48) , then G must contain a k -theta, k -prism, k -pyramid, k -ladder-theta, k -ladder-prism, or k -ladder.Lemmas 17 through 20 are used to prove Lemma 21, which shows that if ( A , B , { x , x , . . . , x k (cid:48) } , { y , y , . . . , y k (cid:48) } ) is a k (cid:48) creature for large enough k (cid:48) , then G [ A ∪ { x , x , . . . , x R ( k (cid:48) ,k (cid:48) ) } ] contains aninduced k -half-theta, k -half-prism, or a k -half-quasi-ladder, ending in { x , x , . . . , x R ( k (cid:48) ,k (cid:48) ) } . Lemma 17.
Let G be a graph that contains a k -creature ( A , B , { x , x , . . . , x k } , { y , y , . . . , y k } )where { x , x , . . . , x k } is an independent set of G . Let A (cid:48) be a minimally connected induced subgraphof G [ A ] such that { x , x , . . . , x k } ⊂ N ( A (cid:48) ) . If A (cid:48) contains a vertex with degree at least R ( d, d ) in A (cid:48) , then G [ A ∪ { x , x , . . . , x k } ] contains a d -half theta or a d -half-prism ending in { x , x , . . . , x k } . Dashed lines represent the option of having an arbitrary length path (possibly of length0). The blue lines used in the k -half-ladder and k -almost-half-ladder graphs represents the option ofeither having or not having that edge, but for each vertex not on the backbone path that is adjacentat least one blue edges, at least one of those blue edges must belong to the graph. roof. Let G be a graph that contains a k -creature ( A , B , { x , x , . . . , x k } , { y , y , . . . , y k } ). Let A (cid:48) be a minimally connected induced subgraph of G [ A ] such that { x , x , . . . , x k } ⊂ N G ( A (cid:48) ). Assume v ∈ A (cid:48) has degree at least R( d, d ) in A (cid:48) . Let v , v , . . . , v R ( d,d ) be distinct neighbors of v in A (cid:48) . Bythe minimality of A (cid:48) , for each v i there must be a vertex x v i such that every path starting from v and ending at x v i with internal vertices contained A (cid:48) must contain v i , since if this does not happenfor some given v i then the connected component of A (cid:48) − v i that contains v would be a properinduced subgraph of A (cid:48) that is connected and whose open neighborhood contains { x , x , . . . , x k } .It follows there must exist induced paths P , P , . . . , P R ( d,d ) such that v i ∈ P i , P i ’s endpoints are v i and x v i , and P i − v i is anti-complete with P j . We then apply Ramsey’s Theorem to the v i ’s geta subset of size d of the P i ’s that along with v form a d -half theta that ends in { x , x , . . . , x k } (ifRamsey’s Theorem provides an independent set of size d ) or a subset of size d of the P i ’s that forma d -half prism that ends in { x , x , . . . , x k } (if Ramsey’s Theorem provides a clique of size d ) andthe result now follows. Lemma 18.
Let G be connected graph with maximum degree d and contains at least d k verticeswith degree greater than 2. Then there exists an induced path of G that contains at least k verticesof degree greater than 2.Proof. Let G be a connected graph with maximum degree d and contains at least d k vertices withdegree greater than 2. Let T be a breadth first search tree of G rooted at some vertex v ∈ G . Wecreate the desired path as follows. Let v be the first descendent of v in T that has degree greaterthan 2 in G ( v could be v ). We begin our path at v . We will grow the path P i = { x , x , . . . x m } where x = v , x j is the parent of x j +1 in T , P i contains at least i vertices of G with degree greaterthan 2 in G , and the subtree of T rooted at x m contains at least d k − i +1 vertices of G with degreegreater than 2 in G .Assume that we have such a path P i = { x , x , . . . x m } , i < k (the vertex v satisfies theconditions of P ). We will show how to attain P i +1 . Since the maximum degree in G is d , x m has at most d children in T , and by assumption the subtree of T rooted at x m has at least d k − i +1 vertices of degree greater than 2 in G , it follows that for at least one child, call it x m +1 , the subtreerooted at x m +1 has at least d k − i vertices of G with degree greater than 2 in G . Now let v i +1 bethe first descendant of x m +1 with degree different from 2 in G ( v i +1 could be x m +1 ) and let P i +1 be the path P i along with the induced path in T from x m +1 to v i +1 . It follows P i +1 satisfies therequired conditions.Hence we can produce a P k that satisfies the conditions stated before, and we can then see that P k is an induced path in G with at least k vertices of degree greater than 2. Lemma 19.
Let G be a graph that contains a k -creature ( A , B , { x , x , . . . , x k } , { y , y , . . . , y k } )where { x , x , . . . , x k } is an independent set. Let A (cid:48) be a minimally connected subgraph of G [ A ] such that { x , x , . . . , x k } ⊂ N ( A (cid:48) ) . If A (cid:48) contains an induced path, P , with at least R ( d, d ) verticesof degree greater than 2 in A (cid:48) , then there is a d -half-quasi-ladder or a d -half-prism in G [ A ∪{ x , x , . . . , x k } ] that ends in { x , x , . . . , x k } .Proof. Let G , A (cid:48) , { x , x , . . . , x k } , and P be as in the statement of the lemma, let v , v , . . . , v R ( d,d ) be vertices of P that have degree greater than 2 in A (cid:48) , and for each v i let v (cid:48) i be a neighbor of v i in A (cid:48) that is not in P . By the minimality of A (cid:48) , for each v (cid:48) i there must exist a vertex x v i such thatevery path from v i to x v i with internal vertices contains in A (cid:48) must contain v (cid:48) i , since if this doesnot happen for some given v (cid:48) i then the component of A (cid:48) − v (cid:48) i that contains v i would be a properinduced subgraph of A (cid:48) that is connected and whose open neighborhood contains { x , x , . . . , x k } .It follows there must exists induced paths P , P , . . . , P R ( d,d ) disjoint from P with internal vertices20ontained in A (cid:48) , P i ’s endpoints are v (cid:48) i and x v i , and P i − v (cid:48) i is anti-complete with P j . We thenapply Ramsey’s Theorem to the v (cid:48) i ’s to get a subset of size d of the P i ’s along with P that forma d -half-quasi-ladder that ends in { x , x , . . . , x k } (if Ramsey’s Theorem provides an independentset of size d ) or a subset of size d of the P i ’s that yield a d -half-prism that ends in { x , x , . . . , x k } (if Ramsey’s Theorem provides a clique of size d ). Lemma 20.
Let G be a graph that contains a k · ( d c +1 + d ) -creature ( A , B , { x , x , . . . , x k · ( d c +1 + d ) } , { y , y , . . . , y k · ( d c +1 + d ) } ). Let A (cid:48) be a minimally connected subgraph of G [ A ] such that { x , x , . . . , x k · ( d c +1 + d ) }⊂ N ( A (cid:48) ) . Assume the max degree in A (cid:48) is d and that A (cid:48) contains less than d c vertices of degreegreater than 2 in A (cid:48) . Then G [ A ∪ { x , x , . . . , x k · ( d c +1 + d ) } ] contains a k -half-quasi-ladder ending in { x , x , . . . , x k · ( d c +1 + d ) } .Proof. Let G , A (cid:48) and { x , x , . . . , x k · ( d c +1 + d ) } be as in the statement of the lemma. Let T be abreadth first search tree of A (cid:48) rooted at some vertex v . Then T is a tree in which every vertexexcept for the root can have at most d − d c + 1 vertices thathave more than one descendent, and the maximum number of decedents any vertex from this setcan have is d . It follows that there are at most d c +1 + d leaves of T , and therefore A (cid:48) is the unionof at most d c +1 + d induced paths in A (cid:48) . Hence, there exists some induced path P in A (cid:48) suchthat P ’s open neighborhood contains at least k vertices in { x , x , . . . , x k · ( d c +1 + d ) } , which gives usa k -half-quasi-ladder ending in { x , x , . . . , x k · ( d c +1 + d ) } . Lemma 21.
Let k (cid:48) = k · R ( k, k ) R ( k,k )+1 + R ( k, k ) , and let G be a graph that contains an R ( k (cid:48) , k (cid:48) ) -creature ( A , B , { x , x , . . . , x R ( k (cid:48) ,k (cid:48) ) } , { y , y , . . . , y R ( k (cid:48) ,k (cid:48) ) } ). Then G [ A ∪{ x , x , . . . , x R ( k (cid:48) ,k (cid:48) ) } ] con-tains an induced k -half-theta, k -half-prism, or a k -half-quasi-ladder, ending in { x , x , . . . , x R ( k (cid:48) ,k (cid:48) ) } .Proof. Let k (cid:48) = k · R ( k, k ) R ( k,k )+1 + R ( k, k ). Assume that G contains a R ( k (cid:48) , k (cid:48) )-creature ( A , B , { x , x , . . . , x R ( k (cid:48) ,k (cid:48) ) } , { y , y , . . . , y R ( k (cid:48) ,k (cid:48) ) } ). Apply Ramsey’s Theorem to { x , x , . . . , x R ( k (cid:48) ,k (cid:48) ) } . IfRamsey’s Theorem returns a clique of size k (cid:48) or more then we have that G [ A ∪{ x , x , . . . , x R ( k (cid:48) ,k (cid:48) ) } ]contains a k -half-prism ending in { x , x , . . . , x R ( k (cid:48) ,k (cid:48) ) } , so we can assume that Ramseys theoremreturns an independent set of size at least k (cid:48) . By relabeling the x i ’s and y i ’s if follows that G containsa k (cid:48) -creature ( A , B , { x , x , . . . , x k (cid:48) } , { y , y , . . . , y k (cid:48) } ) where { x , x , . . . , x k (cid:48) } is an independent set.Let A (cid:48) be a minimally connected induced subgraph of G [ A ] such that { x , x , . . . , x k (cid:48) } ⊂ N ( A (cid:48) ).If A (cid:48) contains a vertex of degree R( k, k ) in A (cid:48) , then by Lemma 17 G [ A ∪ { x , x , . . . , x k (cid:48) } ] containsa k -half-theta ending in { x , x , . . . , x k (cid:48) } . So we may assume max degree of A (cid:48) is R( k, k ).If A (cid:48) contains R( k, k ) R ( k,k ) vertices of degree greater than two, then there is an induced pathof A (cid:48) that contains R( k, k ) vertices of degree greater than two by Lemma 18. Then by Lemma 19 G [ A ∪ { x , x , . . . , x k (cid:48) } ] contains a k -half-quasi-ladder or a k -half-prism ending in { x , x , . . . , x k (cid:48) } .So we may assume that A (cid:48) has maximum degree R( k, k ) and contains fewer than R( k, k ) R ( k,k ) vertices of degree greater than two. It then follows from Lemma 20 that G [ A ∪ { x , x , . . . , x k (cid:48) } ]contains a k -half-quasi-ladder ending in { x , x , . . . , x k (cid:48) } .The next three lemmas show how to clean up a half-quasi-ladder into a half-ladder, half-theta,or theta. Their proofs are similar to those of lemmas 10 12, and 15 respectively, although theconclusions we draw from them are somewhat different. Lemma 22.
Let ( G , S , P , v ) be a tuple where G is a graph, v ∈ G , S ⊂ V ( G ) , and P is aninduced path of G such that ( S ∪ { v } ) and V ( P ) are disjoint. Assume G [ V ( P ) ∪ S ∪ { v } ] does nothave a k -half-theta ending in S , then there is a set X ⊂ S ∪ V ( P ) ∪ { v } of size at most k − suchthat N ( S − N [ X ]) ∩ N ( v ) ∩ V ( P ) = ∅ , and no vertex of S − N [ X ] is neighbors with v . roof. Let G , S , P , and v be as in the statement of this lemma. Number the vertices of P | V ( P ) | so that the vertex numbered i is neighbors with the vertices numbers i − i +1. We nowconsider the following process to build the set desired set X such that N ( S − N [ X ]) ∩ N ( v ) ∩ V ( P ) = ∅ and X ⊂ S ∪ V ( P ) ∪ { v } .We do the following for the first step of the process. Let X = { v } , and let S = { s : s ∈ S − N ( X ) and N ( s ) ∩ N ( v ) ∩ V ( P ) (cid:54) = ∅} (i.e. S is the set of vertices of S − N ( X ) that sharea neighbor with v in P ). Label the vertices of S by the lowest numbered vertex it is neighborswith in V ( P ) ∩ N ( v ). Let s be a highest labeled vertex in S , and let p be s ’s lowest numberedneighbor in N ( v ) ∩ V ( P ). This completes the first step.For the i th step we do the following. Let X i = X i − ∪ { s i − , p i − } , and let S i = S i − − N [ X i ]and label the vertices of S i by the lowest vertex it sees in V ( P ) ∩ N ( v ) (the vertices of S i inherittheir labels from their labels in S i − ). Let s i be a highest labeled vertex in S i and let p i be s i ’slowest neighbor in N ( v ) ∩ V ( P ). Note by how we selected v , s , p , s , p , . . . s i , p i that s a ,1 ≤ a ≤ i , cannot be neighbors with p b if a > b since p b would be in X a and therefore s a wouldnot be in S a , and s a cannot have a neighbor with p b if a < b since that would contradict either p a being s a ’s lowest numbered neighbor in N ( v ) ∩ P or s a being a highest labeled vertex in S a . Hence,we then have that among these vertices s j is only neighbors with p j for 1 ≤ j ≤ i , and v is onlyneighbors with p j for 1 ≤ j ≤ i . p i could be neighbors with p i +1 and/or p i − since they couldbe consecutive vertices on the path P , but p i cannot be neighbors with p j . It follows that the set { v } ∪ { p , p , . . . , p c } ∪ { s , s , . . . , p c } , 2 c ≤ i , forms a c -half-theta in G [ V ( P ) ∪ S { v } ] ending in S . We continue this process until we reach an S j that is empty. By what we noted in the previousparagraph, this process cannot go past the 2 k th step if G [ V ( P ) ∪ S ∪ { v } ] does not contain a k -half-theta ending in S . Set X to be X j . Since S j is empty, it follows N ( S − N [ X ]) ∩ N ( v ) ∩ V ( P ) = ∅ .We also have that no vertex of S − N [ X ] is neighbors with v since v ∈ X and | X | ≤ k − j ≤ k and since the first step adds a single vertex and each step after that only adds twovertices. Lemma 23.
Let ( G, S, P ) be a tuple such that G is a graph, S ⊂ V ( G ) such that S cannot bedominated by k vertices and P is an induced path disjoint from S that dominates S . Assume G [ V ( P ) ∪ S ] does not contain a k -half-theta ending in S . Then there exists a subset S (cid:48) of S of size k such that no vertex of P has more than one neighbor in S (cid:48) .Proof. Let G , S , and P be as in the statement of the lemma. Assume that we have an independentset of vertices vertices S i − of size i − i ≤ k , and a set Z i − of size at most 4 k ( i − S − N [ Z i − ] is neighbors with a vertex in S i − , and any vertex in P thatis neighbor with some vertex in S i − has no other neighbors in S i − nor in S − N [ Z i − ]. We willuse this to produce a set S i of size i and Z i of size at most 4 k i with the same properties. Notethat the empty set satisfies the conditions of S .Let S (cid:48) = S − N [ Z i − ]. Let s be some vertex in S (cid:48) , since i ≤ k and S cannot be dominated by4 k vertices, such an s must exists. We can then apply Lemma 22 using ( G, S (cid:48) , P, s ) and to get aset X of size at most 4 k − S (cid:48) − N [ X ]) ∩ N ( s ) ∩ V ( P ) = ∅ and no vertex of S (cid:48) − N [ X ]is neighbors with s . We then set S i = S i − ∪ { s } and Z i = Z i − ∪ X and we can see these setssatisfies the required properties.Since the empty set satisfies the properties of S and S cannot be dominated by 4 k vertices,we can generate the set S k which has size k and no vertex of P has more than one neighbor in S k . 22 emma 24. Let T be an induced k [2(4 k ) k +1 ] -half-quasi-ladder of a graph G ending in X . Assume T does not have an induced k -half-theta ending in X and assume that G does not contain an induced k -theta. Then T contains a k -half ladder ending in X .Proof. Let G , T , and X be as in the statement of the lemma. Let P be the backbone path of T and P , P , . . . , P k [2(4 k ) k +1 ] be its auxiliary paths, where the endpoints of P i are v i and x i , and the x i ’s are the endpoints of T , so x i ∈ X . Let S = { v , v , . . . , v k [2(4 k ) k +1 ] } . Clearly, if any vertex of P is neighbors with k distinct v i ’s, then T contains a k -half-theta ending in X . It follows that since T does not have a k -half-theta ending in X , the vertices of S cannot be dominated by less than4[2(4 k ) k +1 ] vertices in T . Also, if G [ S ∪ V ( P )] contain a k -half-theta ending in S , then it containsa k -half-theta ending in X , so we can apply Lemma 23 with ( G, P, S ) to get a set S (cid:48) ⊂ S of size2(4 k ) k +1 such that no vertex of P is neighbors with more than one vertex in S (cid:48) . It follows that byonly taking the paths P i such that v i ∈ S (cid:48) , that these P i ’s together with P , form a 2(4 k ) k +1 -half-quasi-ladder where no vertex of P has a neighbor with more than one vertex in any of the P i ’s. Wewill call this 2(4 k ) k +1 -half-quasi-ladder T (cid:48) , we will call its backbone path P (cid:48) so P (cid:48) = P , and wewill call the auxiliary paths P (cid:48) , P (cid:48) , . . . , P (cid:48) k ) k +1 where the endpoints of P (cid:48) i are v (cid:48) i and x (cid:48) i , and the x (cid:48) i ’s are the endpoints of T (cid:48) , so x (cid:48) i ∈ X . We use S (cid:48) as before to denote the set of v (cid:48) i ’s.Now, number the vertices of P (cid:48) | V ( P (cid:48) ) | so that the vertex numbered i is neighborswith the vertices numbers i − i + 1. For a vertex x in P (cid:48) we will use the notation n ( x ) todenote the number it has been given in P (cid:48) . For every s j ∈ S (cid:48) let p j ∈ P (cid:48) be the highest numberedneighbor s j has in P . We now set P = P (cid:48) and S = S (cid:48) . We will consider the following process,where we will try to produce a large independent set in an auxiliary graph related to some P i and S i which we will then use to produce a k -half-ladder. We will show this process cannot go past k iterations if T does not have a k -half-theta ending in X . We will ensure that at the i th step that V ( P i ) ⊂ V ( P (cid:48) ), S i ⊂ S (cid:48) , | S i | ≥ k ) k − i +2 , P i is an induced path, and if s j ∈ S i then p j ∈ P i . Wewill also produce induced subpaths D i of P such that the D i ’s are anti-complete with respect toone another and the vertices of D i will dominate S j if i < j .At the i th step we do as follows. Create an auxiliary directed graph, AU X i , whose vertex set is S i and there is an edge from s a ∈ S i to s b ∈ S i if the following condition holds1. n ( p a ) > n ( p b ) and s a has a neighbor x in P (cid:48) such that n ( x ) < n ( p b )If the maximum in degree of AU X i is at most k | S i | then we stop. If i ≤ k (which we willshow must happen) then since | S i | ≥ k ) k − i +2 this gives an independent set of size at least k byLemma 8. If there is an s j ∈ S i with in degree at least k | S i | then for at least k fraction of thevertices of S i must satisfy (1) playing the role of s a while s j plays the role of s b . Call this set ofvertices S i +1 . If s j ∈ S i with in degree at least k | S i | then we do as follows. Define D i to be thesubpath of P i that is made up of vertices with numbers less than n ( p j ). Set P i +1 to be the verticesof P i with numbers greater than n ( p j ). This concludes the i th step.It can then be seen that V ( P i +1 ) ⊂ V ( P ), S i +1 ⊂ S , | S i +1 | ≥ k ) k − i +1 , P i +1 is an inducedpath, and if s j ∈ S i +1 then p j ∈ P i +1 as required. Furthermore, it can be seen that any of thepreviously D j ’s that have been produced in this process ( j ≤ i ) dominate all vertices of S i +1 . Sincethe D j ’s are disjoint and anti complete, By Lemma 14 then, this process cannot go past the k th iteration without producing a k -theta in G .We conclude there is some step j ≤ k such that the auxiliary graph AU X j has max in-degreeless than k | S j | , and since | S j | ≥ k it therefore has an independent set of size k by Lemma 8. Let S ∗ denote such an independent set.We claim by only taking the paths P (cid:48) i such that v (cid:48) i ∈ S ∗ , that these P (cid:48) i ’s together with P (cid:48) , forma k -half-ladder. Let x, y ∈ S ∗ and let a, b be the highest and lowest numbered neighbors of x in L y has a neighbor c on the induced path of L that has a and b as itsendpoints. If y ’s highest numbered neighbor in L is greater than n ( a ) then y has an edge to x in AU X j . If y ’s highest numbered neighbor in L is less than n ( a ), then x has an edge to y . It followsthat taking the P (cid:48) i such that v (cid:48) i ∈ S ∗ together with P (cid:48) , form a k -half-ladder. Corollary 3.
Let k be a natural number. There exists a natural number k (cid:48) large enough sothat if G is be a graph that contains a k (cid:48) -creature ( A , B , { x , x , . . . , x k (cid:48) } , { y , y , . . . , y k (cid:48) } ), then G [ A ∪ { x , x , . . . , x k (cid:48) } ] contains an induced k -half-theta, k -half-prism, or k -half-ladder ending in { x , x , . . . , x k (cid:48) } or G contains an induced k -theta.Proof. By Lemma 21 there exists a k (cid:48) large enough so that if G contains a k (cid:48) -creature ( A , B , { x , x , . . . , x k (cid:48) } , { y , y , . . . , y k (cid:48) } ) then G [ A ∪ { x , x , . . . , x k (cid:48) } ] contains an induced 4 k [2(4 k ) k +1 ] -half-theta, 4 k [2(4 k ) k +1 ] -half-prism, or a 4 k [2(4 k ) k +1 ] -half-quasi-ladder, ending in { x , x , . . . , x k (cid:48) } .If G [ A ∪ { x , x , . . . , x k (cid:48) } ] contains a 4 k [2(4 k ) k +1 ] -half-theta or a 4 k [2(4 k ) k +1 ] -half-prism end-ing in { x , x , . . . , x k (cid:48) } then we are done. If G [ A ∪ { x , x , . . . , x k (cid:48) } ] contains a 4 k [2(4 k ) k +1 ] -half-quasi-ladder ending in { x , x , . . . , x k (cid:48) } then we may apply Lemma 24 to get that either G [ A ∪ { x , x , . . . , x k (cid:48) } ] contains a k -half-ladder ending in { x , x , . . . , x k (cid:48) } or G contains a k -theta. Lemma 25.
Let k be a natural number. Then there exists a natural number k (cid:48) large enough sothat if G is a graph that contains an k (cid:48) -creature ( A , B , { x , x , . . . , x k (cid:48) } , { y , y , . . . , y k (cid:48) } ), then G contains an induced k -theta, k -prism, k -pyramid, k -ladder-theta, k -ladder-prism, or a k -ladder.Proof. Let k be a natural number. By Corollary 3 there exists a k (cid:48) large enough so that if G is a graph that contains an k (cid:48) -creature ( A , B , { x , x , . . . , x k (cid:48) } , { y , y , . . . , y k (cid:48) } ), then G [ A ∪{ x , x , . . . , x k (cid:48) } ] contains an induced k -half-theta, k -half-prism, or k -half-ladder ending in { x , x , . . . , x k (cid:48) } or G contains an induced k -theta. It then also follows from Corollary 3 there exists a k (cid:48)(cid:48) large enoughso that if G is a graph that contains an k (cid:48)(cid:48) -creature ( A , B , { x , x , . . . , x k (cid:48)(cid:48) } , { y , y , . . . , y k (cid:48)(cid:48) } ), then G [ B ∪ { y , y , . . . , y k (cid:48)(cid:48) } ] contains an induced k (cid:48) -half-theta, k (cid:48) -half-prism, or k (cid:48) -half-quasi-ladder end-ing in { y , y , . . . , y k (cid:48)(cid:48) } or G contains an induced k (cid:48) -theta.So, assume that G is a graph that contains an k (cid:48)(cid:48) -creature ( A , B , { x , x , . . . , x k (cid:48)(cid:48) } , { y , y , . . . , y k (cid:48)(cid:48) } ).If G contains an induced k (cid:48) -theta then we are done, assume that G [ B ∪ { y , y , . . . , y k (cid:48)(cid:48) } ] containsan induced k (cid:48) -half-theta, k (cid:48) -half-prism, or k (cid:48) -half-ladder ending in { y , y , . . . , y k (cid:48)(cid:48) } . By relabel-ing the x i ’s and y i ’s we can then assume that G contains a k (cid:48) creature ( A (cid:48) , B (cid:48) , { x , x , . . . , x k (cid:48) } , { y , y , . . . , y k (cid:48) } ) such that G [ B (cid:48) ∪{ y , y , . . . , y k (cid:48) } ] is a k (cid:48) -half-theta, k (cid:48) -half-prism, or k (cid:48) -half-ladder.Then applying Corollary 3 gives us that G [ A ∪ { x , x , . . . , x k (cid:48) } ] contains an induced k -half-theta, k -half-prism, or k -half-ladder ending in { x , x , . . . , x k } . It follows that G must contain a k -theta,a k -prism, k -pyramid, k -ladder-theta, k -ladder-prism, or a k -ladder.The following two lemmas will be used in Lemma 28 to establish that if F is a family of graphsdefined by a finite number of forbidden induced subgraphs and F allows for at least one of k -thetas, k -prisms, k -pyramids, k -ladder-thetas, or k -ladder-prisms, for arbitrarily large k , then wecan ensure it contains these graphs where their number of vertices only grow linearly with respectto k , and therefore have exponentially many minimal separators. These two lemmas achieve thisby showing that a graph in F has certain paths that are too long, then we can contract part ofthose paths and maintain that the resulting graph remains in F . Lemma 26.
Let G be a graph and let H be a graph with | V ( H ) | ≤ h , where h > . Assume that G contains an induced path P of length at least h where all internal vertices of P have degree 2in G . Then there exists an edge e in G such that if G e contains H as an induced subgraph, then sodoes G . roof. Let G be a graph, let H be a graph with | V ( H ) | ≤ h where h >
5, and let P be an inducedpath of G of length at least 5 h where all internal vertices of P have degree 2, say P = p , p , . . . , p h .Let e be the edge between p (cid:100) h − (cid:101) and p (cid:100) h +12 (cid:101) . Let v denote the new vertex p (cid:100) h − (cid:101) and p (cid:100) h +12 (cid:101) create when e is contracted in G to make G e , and let P (cid:48) be what the path P becomes aftercontracting e in G , so P (cid:48) = p , p , . . . , p (cid:100) h − (cid:101)− , v, p (cid:100) h +12 (cid:101) +1 , . . . , p k . Assume that G e contains H as an induced subgraph. We will show that there exists a set X ⊂ V ( G e ) that induces H such that v / ∈ X . It will then follows that G contains an induced H .Any component of H that is not an induced path can only contain vertices outside of P (cid:48) or withindistance h of either the endpoints of P (cid:48) since all internal vertices of P (cid:48) have degree 2 in G e . For thecomponents of H that are paths, since there are at most h vertices among these components, we canensure that the vertices of X that we use to induce these components either do not belong to P (cid:48) oronly contain vertices from the subpaths p h +2 , p h +3 , . . . , p (cid:100) h − (cid:101)− and p (cid:100) h +12 (cid:101) +1 , p (cid:100) h +12 (cid:101) , . . . , p h − .It follows that v / ∈ X . Lemma 27.
Let G be a graph and let H be a graph with | V ( H ) | ≤ h , where h > . Assume that G contains an induced path P of length h [( h + 1)(5 h ) h +2 + 1] such that the only neighbor the verticesof P might have outside of P is a single vertex v . Then there exists a subpath path P (cid:48) of P suchthat if G P (cid:48) contains H as an induced subgraph, then so does G .Proof. Let G be a graph and let H be a graph with | V ( H ) | ≤ h , where h >
5. Assume that G contains an induced path P of length 5 h [( h + 1)(5 h ) h +2 + 1] such that the only neighbor thevertices of P might have outside of P is a single vertex v . Let a, b be the endpoints of P . Nowdivide P into a sequence of subpaths P , P , . . . , P k each of length at least 2 so that all internalvertices of P i have degree 2 in G , all endpoints of P i are either a vertex of degree 3 or a or b , P has a and one of its endpoints, P k has b as one of its endpoints, and P i shares one of its endpointswith P i +1 (i.e. these are subpaths that whose endpoints are a, b , or the vertices that are neighborswith v and are sequenced going from one end of P to the other). We define a second sequence a , a , . . . a k where a i = | E ( P i ) | . If any a i ≥ h then the result follows from Lemma 26, so we canassume for all i that a i ≤ h . It then follows that k is at least ( h + 1)(5 h ) h +2 + 1, and therefore bythe pigeonhole principle there must be a continuous subsequence of length 2 h + 2 that is repeatedat least h + 2 times, where none of these continuous subsequences overlap with each other. Let S = s , s , . . . , s h +1 be this repeated subsequence. So we have h + 2 sequences for 1 ≤ i ≤ h + 2, A i = a j i , a j i +1 , . . . , a j i +2 h +1 where for 1 ≤ m ≤ h + 2 and c , 0 ≤ c ≤ h + 1, a j m + c = s c and nopart of A m overlaps with some other A n (so | j n − j m | ≥ h + 2) and j m > j n if m > n . Fix thevalues denoted by j m for 1 ≤ m ≤ h + 2.We wish to combine the first half of A with the second half of A by contracting a path in P . Let x be the endpoint of P j + h +1 that it shares with P j + h , and let y be the endpoint P j + h +1 shares with P j + h . Let P (cid:48) be the subpath of P that has x and y as its endpoints. Let w be thevertex that gets created when contracting the path P (cid:48) in G to get G P (cid:48) and let all the subpaths P i of P in G that were not contained in P (cid:48) retain their labels in G P (cid:48) , so P j + h and P j + h +1 share w as an endpoint, and let the a i ’s retain their same meaning as long as P i was not a subpath of P (cid:48) . It follows that G P (cid:48) has h sequences for 3 ≤ i ≤ h + 2, A i = a j i , a j i +1 , . . . , a j i +2 h +1 where for1 ≤ m ≤ h + 2 and c , 0 ≤ c ≤ h + 1, a j m + c = s c and no part of A m overlaps with some other A n (so | j n − j m | ≥ h + 2) and j m > j n if m > n . Furthermore, A and A have now been combined togive A (cid:48) = a j , a j +1 , . . . , a j + h , a j + h +1 , a j + h +2 , . . . , a j +2 h +1 so that a j + c = s c for 0 ≤ c ≤ h and a j + c = s c for h + 1 ≤ c ≤ h + 1. We will show that if there exists a set X ⊂ V ( G P (cid:48) ) that induces H in G P (cid:48) then we can require w / ∈ X . The result then follows since if w / ∈ X then the vertices thatcorrespond to X in G induced an H in G . 25o, assume X ⊂ V ( G P (cid:48) ) and induces H . If w / ∈ X then we are done, so assume w ∈ X (cid:48) forsome connected component X (cid:48) of X . For i with 3 ≤ i ≤ h + 1, let P ∗ i denote the path induced by V ( P j i ) , V ( P j i +1 ) , . . . , V ( P j i +2 h +1 ) in G P (cid:48) , so P ∗ i is the path that naturally corresponds to S i , and let P ∗ denote the path induced by V ( P j ) , V ( P j +1 ) , . . . , V ( P j + h ) , V ( P j + h +1 ) , V ( P j + h +2 ) , . . . , V ( P j +2 h +1 ),so P ∗ naturally corresponds with A (cid:48) . Then since X (cid:48) has at most h vertices there is at least one P ∗ i that contains no vertex of X and since X (cid:48) is connected and contains w , all vertices of X (cid:48) ∩ P must be completely contained in V ( P ∗ ) since w is at least distance h from either endpoint of P ∗ .It follows that we can replace the vertices of X (cid:48) ∩ P , which must be completely contained in theinteral vertices of P ∗ , with the corresponding vertices in a P ∗ i that contains no vertices of X andstill maintain that the vertices of X induce H . Now w / ∈ X and the result then follows. Lemma 28.
Let F be a family of graphs determined by a finite number of forbidden inducedsubgraphs. Then if F does not forbid all k -thetas, k -prisms, k -pyramids, k -ladder-thetas, k -ladder-prisms, and k -ladders for arbitrarily large k , then F is feral.Proof. Let F be a family of graphs determined by a finite number of forbidden induced subgraphs,and let H be a set of forbidden subgraphs that define F . Let let h > H ∈ H , | V ( H ) | ≤ h . First assume that F allows for either k -thetas k -prisms, or k -pyramidsfor arbitrarily large k . Then by Lemma 26 we can ensure that all paths with internal vertices allhaving degree 2 of the k -thetas k -prisms, or k -pyramids are at most 5 h (we keep on contractingthe appropriate edges given by Lemma 26 until no path where all internal vertices have degree 2have length more than 5 h ) and therefore F contains a k -theta k -prism, or k -pyramid with at most5 h · k vertices. Since a k -theta, k -prism, or k -pyramid must have at least 2 k minimal separators, itfollows that there exists a c > N ther exists a G ∈ F suchthat | V ( G ) | = n > N and the number of minimal separators in G is at least c n .Now assume that F allows for k -ladder-thetas or k -ladder-prisms for arbitrarily large k . Every k -ladder-theta and k -ladder-prism contains a k -half-ladder and by Lemma 26 we can ensure that allpaths with internal vertices all having degree 2 of the k -ladder-theta or k -ladder-prism are at most5 h and by Lemma 27 we can ensure that the backbone path of the corresponding k -half-ladderhas length at most [5 h ( h + 1)(5 h ) h +1 + 1] · k by contracting the appropriate edges and paths ifnecessary while still guaranteeing the resulting graph belongs to F (Lemma 27 gives us that ifthere is a subpath of length over [5 h ( h + 1)(5 h ) h +1 + 1] of the backbone path that only has oneneighbor outside of the backbone path, there there exists a subpath of the backbone path thatwe can contract and still maintain that the resulting graph is a k -ladder-theta or k -ladder-prismcontained in F ). Since k -ladder-thetas and k -ladder-prisms have at least 2 k minimal separatorsit follows that there exists a contains c > N there exists a G ∈ F such that the number of minimal separators in G is at least c n . It follows that F is feral.The following lemma shows why it is necessary to forbid k -paw and k -claw graphs for a familyof graphs defined by a finite number of forbidden induces subgraphs to be strongly-quasi-tame.Figure 5 gives a picture of the two graphs constructed in the following lemma. Lemma 29.
Let F be a family of graphs determined by a finite number of forbidden inducedsubgraphs. Then if F does not forbid k -claws and k -paws for some natural number k , then F isferal.Proof. Let F be a family of graphs determined by a finite number of forbidden induced subgraphs,and let H be a set of forbidden subgraphs that define F . Let h > H ∈ H , | V ( H ) | ≤ h . First we assume that F allows k -claw for arbitrarily large k . We willconstruct a graph with many minimal separators. Assume that we have two set of 2 c − The two graphs in this figure are small versions of the constructions of the graphs givenin Lemma 29, explicit vertices are omitted in this graph. The left side graph is the constructionprovided when when the k -claw is not forbidden for arbitrarily large k . The right hand side graphis the construction provided when when the k -paw is not forbidden for arbitrarily large k . C , C , . . . C c , and C , C , . . . C c where in both sets each long claw has arm length h . We labelthe leaves of C i as a i , b i , c i and we label the endpoints of C i as a i , b i , c i . Then for 1 ≤ i ≤ c − − a i to b i , a i +1 to c i , a i to b i , and a i +1 to c i . Furthermore, for 2 c − ≤ i ≤ c − b i and b i and between c i and c i . Note that any collection of b j i i and c (cid:96) i i with2 c − ≤ i ≤ c − j i , (cid:96) i = 1 or 2 is a minimal separator, so there are at least 2 c minimalseparators in this construction. Since the arm length of each long-claw is h , the total number ofvertices in this construction is less than 3 h · c +1 .If F allows for k -claws, then forest of paths and subdivided claws cannot be forbidden in F , andit can be seen that any induced subgraph of size at most h of the construction just given is a forestof paths and subdivided claws (i.e. three anti-complete paths where one endpoint of each path areglued together). It follows that this construction must belong to F and since this construction hasat least 2 c minimal separators and less than 3 h · c +1 vertices, the statement of the lemma followfor the case where k -claw graphs for arbitrarily large k are not forbidden.Now we assume that F allows k -paw graphs for arbitrarily large k . The construction and analysiswe make in this case is nearly identical to the k -claw case. We present it here for completeness.Assume that we have two set of 2 c − C , C , . . . C c , and C , C , . . . C c where in bothsets each long-paw has arm length h . We label the endpoints of C i as a i , b i , c i and we label theendpoints of C i as a i , b i , c i . Then for 1 ≤ i ≤ c − − a i to b i , a i +1 to c i , a i to b i , and a i +1 to c i . Lastly, for 2 c − ≤ i ≤ c − a i and a i and between b i and b i . Note that any collection of b j i i and c (cid:96) i i with 2 c − ≤ i ≤ c − j i , (cid:96) i = 1 or 2 is a minimalseparator, so there are at least 2 c minimal separators in this construction. Since the arm lengthof each long-claw is h , the total number of vertices in this construction is less than 3 h · c +1 .Since F allows for k -paws, a forest of paths and subdivided paws cannot be forbidden in F ,and it can be seen that any induced subgraph of size at most h of the construction just given is aforest of paths and subdivided paws. It follows that this construction must belong to F and sincethis construction has at least 2 c minimal separators and less than 3 h · c +1 vertices, the statementof the lemma follows for the case where k -paw graphs for arbitrarily large k are not forbidden.We are now ready to prove Theorem 2 Proof of Theorem 2.
Let F be a family of graphs defined by a finite number of forbidden inducedsubgraphs. It follows from Lemmas 28 and 29 that if F allows for any k -thetas, k -prisms, k -27yramids, k -ladder-thetas, k -ladder-prisms, k -claws, or k -paws for arbitrarily large k , F is feral.Now assume that there exists a natural number k such that F forbids k -thetas, k -prisms, k -pyramids, k -ladder-thetas, k -ladder-prisms, k -claws, and k -paws. Observe that there exists a k (cid:48) large enough so that if G contains an induced k (cid:48) -ladder, then G contains an induced k -claw or k -paw graph, therefore F forbids k (cid:48) -ladders. It then follows from Lemma 25 there exists a k (cid:48)(cid:48) such that no G ∈ F can contain a k (cid:48)(cid:48) -creature, where the minimum value of k (cid:48)(cid:48) is a function of k .Furthermore, it is clear that there exists a k (cid:48)(cid:48)(cid:48) large enough so that if G contains a k (cid:48)(cid:48)(cid:48) -skinny-ladderas an induced minor, then G contains a k -claw or a k -paw as an induced subgraph. Hence F forbids k (cid:48)(cid:48)(cid:48) -skinny-ladders as an induced minor. It then follows from Theorem 1 that there is a function f : N → N such that for all G ∈ F the number of minimal separators of G is at most n f ( k ) log( n ) .Hence F is tame. Here we present a proof of Theorem 3 which is based on an easy application of Corollary 2. Wewill need the following lemma in order to apply Corollary 2.
Lemma 30.
Let G be a C ≥ r -free graph and assume G does not contain a k -creature. Then everyminimal separator, S , can be dominated by r · k vertices of G not in S .Proof. Let G be a C ≥ r -free graph and assume G does not contain a k -creature. Assume for acontradiction that there exists a minimal separator, S , of G such that S cannot be dominated by r · k vertices in G and not in S . Let H be an S -full component of G − S , then by Lemma 9, S isdominated a subset of H that is the union of k induced paths in H . It follows there must existssome induced path P in H such that S P = N ( P ) ∩ S cannot be dominated by r vertices in P .There then exists a subpath P (cid:48) of P such that there are vertices a, b ∈ S P that have no neighborin P (cid:48) , both component of P − P (cid:48) have vertices that are neighbors with a and/or b . It follows thatwe can extend the path P (cid:48) to have endpoints x a and x b such that the only neighbors of a in P (cid:48) is x a and possible x b and the only neighbors of b in P (cid:48) is x b and possibly x a . If x a and x b areboth neighbors with a then P (cid:48) and a form a cycle of length r , and if x a and x b are both neighborswith b then P (cid:48) and b form a cycle of length r so assume neither of these cases occur. If a and b are neighbors then P (cid:48) a , b make a cycle of length more than r . Else, there is an induced path, T between a and b with all of its internal vertices contained in some S -full component other than H .It follows that P (cid:48) , and T makes a cycle of length more than r , a contradiction. Proof of Theorem 3.
Let G be a C ≥ k -free graph that is k -theta, k -prism, and k -pyramid free. Since G is C ≥ k -free this implies that G is also k -ladder-theta, k -ladder-prism, and k -ladder free. Lemma25 then implies that there exists a function f : N → N (independent of the choice of k or G )such that G is f ( k )-creature-free. Lemma 30 gives that every minimal separator S of G can bedominated by kf ( k ) vertices not in S . Hence, by Corollary 2 G has at most | V ( G ) | ( kf ( k ) ) +2 kf ( k ) minimal separators. It follows that the family of graphs that are C ≥ k -free, k -theta, k -prism, and k -pyramid free is tame. Here we present a proof of Theorems 4 and 5 which are based on an easy application of Corollary2. We will need the following lemma in order to apply Corollary 2.28 emma 31.
Let k (cid:48) = k ) k +1 ] . If G is k -creature free, G does not contain a k -skinny-ladder asan induced minor, and no minimal separator of G contains a clique of size k , then every minimalseparator S of G can be dominated by at most ( k (cid:48) ) k +1 vertices of G − S .Proof. Let k (cid:48) , k , and G be as in the statement of the lemma. Let G (cid:48) be an induced subgraph of G and let S (cid:48) be a minimal separator of G (cid:48) . Then G (cid:48) must be k -creature free and k -ladder free, so itfollow from Lemma 15 that S (cid:48) can be dominated by k (cid:48) vertices of G (cid:48) − S (cid:48) .We will produce a set of ( k (cid:48) ) k +1 vertices of G − S that dominate S by considering the followingrecursive algorithm. The input to the algorithm is ( G (cid:48) , S (cid:48) ) where G (cid:48) is a subgraph of G and S (cid:48) is aminimal separator of G (cid:48) , and the algorithm returns a set of vertices which will be described shortly.The algorithm finds two vertex sets A and B such that | A | + | B | ≤ k (cid:48) , A ⊂ V ( G (cid:48) ), B ⊂ S (cid:48) , and A ∪ B dominate S (cid:48) (such a set must exists by what was established in the previous paragraph).Let B (cid:48) be a set of vertices in G (cid:48) − S (cid:48) such that | B (cid:48) | ≤ | B | and B (cid:48) dominates B . For each b ∈ B we recursively call the algorithm on ( G (cid:48) − ( S (cid:48) − [ S (cid:48) ∩ N ( b )]) , S (cid:48) ∩ N ( b )) (note that S (cid:48) ∩ N ( b ) is aminimal separator of G (cid:48) − ( S (cid:48) − [ S (cid:48) ∩ N ( b )])). Let X be the union of the sets returned by eachrecursive call. Then algorithm then returns X ∪ A ∪ B (cid:48) .If we initially call this algorithm on ( G, S ) for some minimal separator S of G , then it is clearthat the set this algorithm returns is a subset of vertices of G − S that dominate S . We can alsosee the depth of this recursive algorithm cannot go past k without producing a clique of size k in S since the minimal separator we recursively call this algorithm on is always dominated by theopen neighborhood of some vertex v of S . So, the depth of the recursion tree is at most k − k (cid:48) children since | B | ≤ k (cid:48) . It follows that since each recursive call of thealgorithm adds at most k (cid:48) vertices to the set it returns, the size of the final returned set cannotexceed k (cid:48) · k (cid:48) k Proof of Theorem 4.
Let G be a graph that is k -creature free and does not contain a k -skinny-ladder as an induced minor, and furthermore assume that no minimal separator of G has a cliqueof size k . By Lemma 31 there exists a function f : N → N such that all minimal separators, S , ofany graph that is k -creature free, does not contain a k -skinny-ladder as an induced minor, and hasno minimal separator that contains a clique of size k , can be bounded by f ( k ) vertices outside of S .It then follows from Corollary 2 that G has at most | V ( G ) | f ( k ) +2 f ( k ) minimal separators. Hence,the family of graphs that are k -creature free, do not contain a k -skinny-ladder as an induced minor,and have no minimal separator has a clique of size k is tame. Proof of Theorem 5.
Let F be a family of graphs defined by a finite number of forbidden inducedsubgraphs. Assume that F forbids the complete graph on k vertices for some natural number k .It follows from Lemmas 28 and 29 that if F allows for any k (cid:48) -thetas, k (cid:48) -ladder-thetas, k (cid:48) -claws, or k (cid:48) -paws for arbitrarily large k (cid:48) , then F is feral.Now assume that for some integer k that F forbids k -thetas, k -ladder-thetas, k -claws, and k -claws. Since F forbids k -cliques as well, it follows that F forbids k -prisms, k -pyramids, and k -ladder-prisms. Observe that there exists a k (cid:48) large enough so that if G contains an induced k (cid:48) -ladder, then G contains an induced k -claw or k -paw, therefore G does not contain a k (cid:48) -ladder.It follows from Lemma 25 there exists a k (cid:48)(cid:48) such that no G ∈ F can contain a k (cid:48)(cid:48) -creature, wherethe minimum value of k (cid:48)(cid:48) is a function of k . Furthermore, it is clear that there exists a k (cid:48)(cid:48)(cid:48) largeenough so that if G contains a k (cid:48)(cid:48)(cid:48) -skinny-ladder as an induced minor, then G contains a k -claw ora k -paw as an induced subgraph. Hence F forbids k (cid:48)(cid:48)(cid:48) -skinny-ladders as an induced minor. Now, ifno graph of F contains a minimal separator with a clique of size k , then it then follows by Lemma31 there exists a function f : N → N such that for all G ∈ F it holds that all minimal separators S G can be bounded by f ( k ) vertices in G − S . It then follows from Corollary 2 that for all G ∈ F has at most | V ( G ) | f ( k ) +2 f ( k ) minimal separators. Therefore F is tame. In this paper we disproved a conjecture of Abrishami et al. [ACD +
20] that for any natural number k , the family of graphs that exclude k -creatures is tame. On the other hand, we proved a weakenedform of the conjecture, that every family of graphs that excludes k -creatures and also excludes k -skinny ladders as induced minors is strongly-quasi-tame. This led to a complete classification ofgraph families defined by a finite number of forbidden induced subgraphs into strongly-quasi-tameand feral, substantially generalizing the main result of Milaniˇc and Pivaˇc [MP19]. The tools wedevelop on the way to prove our main results yield with some additional effort polynomial upperbounds instead of quasi-polynomial, proving tameness instead of strong quasi-tameness, for twointeresting special cases. In particular we show that the conjecture of Abrishami et al. [ACD + C ≥ r -free graphs for every integer r , as well as for K r -free graphs excluding an r -skinnyladder for every integer r . The first of these results generalizes work of Chudnovsky et al. [CPPT19],who proved that C ≥ -free, k -creature free graphs are tame,Although Theorems 1 and 2 provide a strongly-quasi-tame bound we have no examples of non-tame families that exclude k -creatures and k -skinny ladders for some k . We conjecture that theseclasses of graphs are actually tame. Conjecture 2.
For every natural number k , the family of graphs that are k -creature free and donot contain a k -skinny-ladder as an induced minor is tame. Conjecture 2, if true, put together with the proof of Theorem 2 would lead to the followingclassification of hereditary families defined by a finite set of forbidden induced subgraphs.
Conjecture 3.
Let F be a graph family defined by a finite number of forbidden induced subgraphs.If there exists a natural number k such that F forbids all k -theta, k -prism, k -pyramid, k -ladder-theta, k -ladder-prism, k -claw, and k -paw graphs, then F is tame. Otherwise F is feral. We remark that Conjecture 2 implies Conjecture 3, but not the other way around. In particularConjecture 3 might be easier to prove.We have so far been unsuccessful in identifying other counterexamples to Conjecture 1 thatlook “substantially different” from the k -twisted ladders constructed in Section 3. For this reasonit is tempting to conjecture that at least for induced minor closed classes, a ”clean” classificationof all classes into tame or feral is possible. Conjecture 4.
Every induced-minor-closed class F is either tame or feral.
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