On the tree-width of even-hole-free graphs
Pierre Aboulker, Isolde Adler, Eun Jung Kim, Ni Luh Dewi Sintiari, Nicolas Trotignon
OOn the tree-width of even-hole-free graphs
Pierre Aboulker ∗ Isolde Adler † Eun Jung Kim ‡ Ni Luh Dewi Sintiari § Nicolas Trotignon § Abstract
The class of all even-hole-free graphs has unbounded tree-width, asit contains all complete graphs. Recently, a class of (even-hole, K )-freegraphs was constructed, that still has unbounded tree-width [Sintiariand Trotignon, 2019]. The class has unbounded degree and containsarbitrarily large clique-minors. We ask whether this is necessary.We prove that for every graph G , if G excludes a fixed graph H as a minor, then G either has small tree-width, or G contains a largewall or the line graph of a large wall as induced subgraph. This canbe seen as a strengthening of Robertson and Seymour’s excluded gridtheorem for the case of minor-free graphs. Our theorem implies thatevery class of even-hole-free graphs excluding a fixed graph as a minorhas bounded tree-width. In fact, our theorem applies to a more generalclass: (theta, prism)-free graphs. This implies the known result thatplanar even hole-free graph have bounded tree-width [da Silva andLinhares Sales, Discrete Applied Mathematics 2010].We conjecture that even-hole-free graphs of bounded degree havebounded tree-width. If true, this would mean that even-hole-freenessis testable in the bounded-degree graph model of property testing. Weprove the conjecture for subcubic graphs and we give a bound on the ∗ DIENS, ´Ecole normale sup´erieure, CNRS, PSL University, Paris, France. Email: [email protected] by grant ANR-19-CE48-0016 from the French National Research Agency(ANR). † School of Computing, University of Leeds, Leeds, LS2 9JT, UK. Email:[email protected] research was partly carried out during a visit to Universit´e Paris Dauphine, fundedby LAMSADE. ‡ Universit´e Paris-Dauphine, PSL University, CNRS UMR7243, LAMSADE, Paris,France. Email: [email protected] by the grant from the French National Agency under JCJC program (ASSK:ANR-18-CE40-0025-01). § Univ Lyon, EnsL, UCBL, CNRS, LIP, F-69342, LYON Cedex 07, France. Email: { nicolas.trotignon,ni-luh-dewi.sintiari } @ens-lyon.fr.Partially supported by the LABEX MILYON (ANR-10-LABX-0070) of Universit´e deLyon, within the program ‘Investissements d’Avenir’ (ANR-11-IDEX-0007) operated bythe French National Research Agency (ANR). a r X i v : . [ c s . D M ] A ug ree-width of the class of (even hole, pyramid)-free graphs of degree atmost 4. Keywords
Even-hole-free graphs, grid theorem, tree-width, bounded-degree graphs, property testing
Theory of computation → Math-ematics of computing → Discrete mathematics → Graph Theory; Theory ofcomputation → Design and analysis of algorithms → Streaming, sublinearand near linear time algorithms
Here, all graphs are simple and undirected. A hole in a graph is an inducedcycle of length at least 4. It is even or odd according to the parity of its length , that is the number of its edges. We say that a graph G contains a graph H if some induced subgraph of G is isomorphic to H . A graph is H -free if it does not contain H . When H is a set of graphs, G is H -free if G contains no graph of H . A graph is therefore even-hole-free if it does notcontain an even hole.Even-hole-free graphs were the object of much attention, see for instancethe survey [24]. However, many questions about them remain unanswered,such as the existence of a polynomial time algorithm to color them, or to finda maximum stable set. In fact, to the best of our knowledge, no problemthat is polynomial time solvable for chordal graphs is known to be NP-hard for even-hole-free graphs (where a chordal graph is a hole-free graph).Despite the existence of several decomposition theorems or structural prop-erties (see [24]), no structure theorem is known for even-hole-free graphs.In addition, motivated by the question whether even-hole-freeness istestable in the bounded degree model of property testing, the structureof even-hole-free graphs of bounded maximum degree is of interest. If even-hole-free graphs of bounded degree have bounded tree-width, it would implytestability in the bounded degree model, because even-hole-freeness is ex-pressible in monadic second-order logic with modulo counting (CMSO) andCMSO is testable on bounded tree-width [1]. We will discuss this in greaterdetail below. Let us first provide some more background. Background.
We begin by recalling known definitions and results abouttree-width. The clique number of a graph G , denoted by ω ( G ), is the max-imum number of pairwise adjacent vertices in G . The tree-width of a graph G is the minimum of ω ( J ) − J such that G is asubgraph of J . The tree-width can be seen as a measure of the structuraltameness of a graph: the smaller the tree-width, the more ‘tree-like’ thegraph. A celebrated result [9] asserts that many problems (including graph2oloring or finding a maximum stable set) can be solved in polynomial timewhen restricted to graphs of bounded tree-width. However, many graphswith in some sense a simple structure have large tree-width. For instancethe complete graph on n vertices, that we denote by K n , has tree-width n −
1. A graph H is a minor of a graph G , if H can be obtained from asubgraph of G by contracting edges. Tree-width is monotone under takingminors in the sense that if H is a minor of G , then the tree-width of H isless than or equal to the tree-width of G . It follows that graphs that contain K n as a minor have tree-width at least n −
1. The converse is not true: gridshave arbitrarily large tree-width but they do not contain K as a minor.The class of all even-hole-free graphs trivially has unbounded tree-width,as is contains all complete graphs. Also, chordal graphs form a well studiedsubclass of even-hole-free graphs of unbounded tree-width. However, even-hole-free graphs with no triangle have bounded tree-width [7]. This leadsto asking whether even-hole-free graphs of bounded clique number havebounded tree-width – a question that is first asked and motivated in [6].This was answered negatively in [22], where (even hole, K )-free graphs ofarbitrarily large tree-width are described. However, the construction usesvertices of large degree and a large clique minor to increase the tree-width,and it seems natural to ask whether this is necessary.It is known that planar even-hole-free graphs have bounded tree-width [21],and planar graphs do not contain K (cid:96) as a minor for (cid:96) ≥
5. Besides that,it is known that an upper bound on the length of the largest induced cycleimplies an upper bound on the tree-width for graphs of bounded maximumdegree [4].
Our contributions.
The results explained above suggest the followingtwo conjectures.
Conjecture 1.
There is a function f : N → N such that every even-hole-freegraph not containing K (cid:96) as a minor has tree-width at most f ( (cid:96) ) . Conjecture 2.
There is a function f : N → N such that every even-hole-freegraph of degree at most d has tree-width at most f ( d ) . In this paper we prove Conjecture 1 (cf. Section 3). Indeed, we provethe following stronger result, which implies Conjecture 1.
Theorem 1.1 (Induced grid theorem for minor-free graphs) . For everygraph H there is a function f H : N → N such that every H -minor-free graphof tree-width at least f H ( k ) contains a ( k × k )-wall or the line graph of achordless ( k × k ) -wall as an induced subgraph. Here a wall is a (possibly subdivided) hexagonal grid (cf. Section 3).Slightly more generally, Theorem 1.1 implies that (theta, prism)-free graphs (to be defined in Section 2) exluding a fixed minor have bounded3ree-width. Note that (theta, prism)-free graphs form a superclass of even-hole-free graphs.Our theorem can be seen as ‘induced’ version on minor-free graphs classesof the following famous theorem.
Theorem 1.2 (Robertson and Seymour [20]) . There is a function f : N → N such that every graph of tree-width at least f ( k ) contains a ( k × k ) -wall asa subgraph. Theorem 1.2 cannot be strengthened to finding walls as induced sub-graphs in general, because the complete graph K n has tree-width n − × K )-free graphsof unbounded tree-width in [22], vertices of large degree and large cliqueminors are needed, we make the following conjecture. Conjecture 3.
For every d ∈ N there is a function f d : N → N such thatevery graph with degree at most d and tree-width at least f d ( k ) contains a ( k × k )-wall or the line graph of a ( k × k ) -wall as an induced subgraph. Conjecture 3 implies Conjecture 2. Conjecture 3 is wide open, and ourresults can be seen as a step in the direction of a proof.For Conjecture 2 (that is trivial for l ≤ l = 3by providing a full structural description of subcubic even-hole-free graphs(a graph is subcubic if it does not contain a vertex of degree more than 3).In fact, all these results apply to (theta, prism)-free graphs (cf. Section 2for the details). We also prove a weakening of Conjecture 2 for l = 4 (cf.Section 5). Motivation from property testing.
Our other source of motivation forstudying even-hole-free graphs of bounded degree stems from the questionwhether even-hole-freeness is testable in the bounded degree graph model.Motivated by the growing need of highly efficient algorithms, in particu-lar when the inputs are huge, property testing aims at devising sublineartime algorithms. Property testing algorithms (simply called testers ) solve arelaxed version of decision problems, they are randomised, and they comewith a small controllable error probability.Since the input cannot be read even once in sublinear time, testers havelocal access to the input graph only. The bounded degree graph model as-sumes a fixed upper bound d on the degree of all graphs, and the testersproceed by sampling a constant number of vertices of the input graph and ex-ploring their local (constant radius) neighborhoods. A property P is testable ,if there is an (cid:15) -tester for P , for every fixed small (cid:15) >
0. For an input graph4 , an (cid:15) -tester determines, with probability at least 2 / G has property P , or G is (cid:15) -far from having property P . Here a property is simply an isomorphism closed class of graphs. A graph G is (cid:15) -close to P , if there is a graph G (cid:48) ∈ P on the same number n of vertices as G , suchthat G and G (cid:48) can be made isomorphic by at most (cid:15)dn edge modifications(deletions or insertions) in G or G (cid:48) , and otherwise, G is (cid:15) -far from P . Thenumber of vertices explored in the input graph is called the query complexity of the tester. The model requires testers to have constant query complexity.Properties that are known to be testable in the bounded degree graphmodel include subgraph-freeness (for a fixed subgraph), k -edge connectivity,cycle-freeness, being Eulerian, degree-regularity [14], bounded tree-widthand minor-freeness [3, 15, 18], hyperfinite properties [19], k -vertex connec-tivity [25, 12], and subdivision-freeness [17]. Properties that are not testablein the bounded degree model include bipartiteness, 3-colorablilty, expansionproperties, and k -clusterability, cf. [13].Since the testers can only explore a constant number of constant radiusneighborhoods in the input graph, intuitively, properties that are testableshould have some form of ‘local’ nature. Hence it might seem unlikely thatproperties like Hamiltonicity and even-hole-freeness are testable, due to thelarge cycles involved. Indeed, it can be shown that there is no one-sidederror tester for Hamiltonicity (i. e. there is no tester that always accepts yes-instances) with constant query complexity. More precisely, every one-sidedtester has query complexity at least Ω( n ) [16].Perhaps surprisingly, our results suggest a different picture for even-hole-freeness. It is known that on graphs of bounded degree and boundedtree-width, every property that can be expressed in monadic second-orderlogic with counting (CMSO), is testable with constant query complexityand polylogarithmic running time [1]. (Here polylogarithmic in n meansbounded by a polynomial in log n .) It is straightforward to see that even-hole-freeness is expressible in CMSO. This can be done by expressing thatthere is no set X of edges that form an induced hole, where | X | is even(cf. eg. [10] for more details on CMSO expressibility). Together with thefact that bounded tree-width is testable, this implies that if Conjecture 2is true, then even-hole-freeness is testable with constant query complexityand polylogarithmic running time. (This is done by first testing for boundedtree-width and, if the answer is positive, testing for even-hole-freeness usingCMSO testability.) Our results in Section 4 imply the following.
Theorem 1.3.
On subcubic graphs, even-hole-freeness is testable with con-stant query complexity and polylogarithmic running time.
Structure of the paper.
We start with fixing notation in Section 2. Sec-tion 3 contains the proof of the induced grid theorem for minor-free graph5lasses, and the proof of Conjecture 1. Section 4 contains the structure the-orem for subcubic (theta, prism)-free graphs and the proof of Conjecture 2for d = 3. Recall that (theta, prism)-free graphs are defined in the nextsection, and it is a superclass of even-hole-free graphs. In Section 5 we pro-vide a structure theorem for (even hole, pyramid)-free graphs of maximumdegree 4, and we derive a bound on the tree-width of this class (pyramidswill be defined in the next section). In Section 6 we give ideas suggestingthat a structure theorem for even-hole-free graphs with maximum degree 4might exist, and if so, should imply bounded tree-width. We let N denote the set of natural numbers including 0. We use X (cid:116) Y instead of X ∪ Y if X ∩ Y = ∅ . We use (0 ,
1] to denote the real interval thatexcludes 0 and includes 1. For any n ∈ N , n ≥
1, let [ n ] := { , . . . , n } .An (undirected) graph is a pair G = ( V ( G ) , E ( G )), consisting of a set V ( G ), the set of vertices of G , and a set E ( G ) of edges of G , where an edgeis a two-vertex subset of V ( G ). A graph H is a subgraph of a graph G ,if V ( H ) ⊆ V ( G ) and E ( H ) ⊆ E ( G ). For a set X ⊆ V ( G ) the subgraph induced by X in G is the subgraph G [ X ] of G with vertex set X , suchthat e ∈ E ( G [ X ]) iff e ∈ E ( G ) and e ⊆ X . A graph H is an inducedsubgraph of G , if H = G [ X ] for some X ⊆ V ( G ). For a set S ⊆ V ( G ) we let G \ S := G [ V ( G ) \ S ] and if S = { v } is a singleton set, then we write G \ v instead of G \ { v } .A path in G is a sequence P of distinct vertices p . . . p n , where for i, j ∈ { , . . . , n } , p i p j ∈ E ( G ) if and only if | i − j | = 1. For two vertices p i , p j ∈ V ( P ) with j > i , the path p i p i +1 . . . p j is a subpath of P that isdenoted by p i P p j . The subpath p . . . p n − is called the interior of P . Thevertices p , p n are the ends of the path, and the vertices in the interior of P are called the internal vertices of P . A cycle is defined similarly, withthe additional properties that n ≥ p = p n . The length of a path P is the number of edges of P . The length of cycle is defined similarly. Let G be a graph. For vertices u, v ∈ V ( G ), the distance between u and v in G , denoted by dist G ( u, v ), is the length of a shortest path from u to v , if apath exists, and ∞ otherwise. For two subsets X, Y ⊆ V ( G ), the distance between X and Y is min { dist G ( x, y ) | x ∈ X, y ∈ Y } . For v ∈ V ( G ), we callthe set N Gr ( v ) := { w ∈ V ( G ) | dist G ( v, w ) ≤ r } the r -neighborhood of v (in G ). The degree of a vertex v in G is defined as deg G ( v ) := |{ u ∈ V ( G ) |{ u, v } ∈ E ( G ) }| . The degree of G , deg( G ), is the maximum degree over allvertices of G . By C d we denote the class of all graphs of degree at most d ∈ N .The line graph of a graph G is the graph L ( G ), with V ( L ( G )) = E ( G )6nd two vertices of L ( G ) are adjacent, if their corresponding edges are in-cident in G . An edge e ∈ E ( G ) is a chord of cycle C , if the endpoints of e are vertices of C that are not adjacent on C . A hole is a chordless cycleof length at least 4. A clique in G is a set X ⊆ V ( G ) of vertices such that { v, w } ∈ E ( G ) for every pair v, w ∈ X with v (cid:54) = w . A graph K is complete ,if V ( K ) is a clique in K . We use K k to denote the complete graph on k vertices. For disjoint sets A, B ⊆ V ( G ), we say that A is anticomplete to B if no edges are present between A and B in G .A pyramid is a graph made of three chordless paths P = x . . . a , P = x . . . b , P = x . . . c , each of length at least 1, two of which have length atleast 2, internally vertex-disjoint, and such that abc is a triangle and noedges exist between the paths except those of the triangle and the threeedges incident to x . The vertex x is called the apex of the pyramid.A prism is a graph made of three vertex-disjoint chordless paths P = a . . . a (cid:48) , P = b . . . b (cid:48) , P = c . . . c (cid:48) of length at least 1, such that abc and a (cid:48) b (cid:48) c (cid:48) are triangles and no edges exist between the paths except those of the twotriangles.A theta is a graph made of three internally vertex-disjoint chordless paths P = a . . . b , P = a . . . b , P = a . . . b of length at least 2 and such that noedges exist between the paths except the three edges incident to a and thethree edges incident to b .A wheel is a graph formed from a hole H together with a vertex x thathas at least three neighbors in the hole. Such a hole H is called the rim ,and such a vertex x is called the center of the wheel. We denote by ( H, x ),the wheel with rim H and center x .Figure 1: Pyramid, prism, theta, and wheel (dashed lines represent paths)Theta and prism are relevant in this work, because of the following well-known lemma. The following lemma clearly implies that (theta, prism)-freegraphs form a superclass of even-hole-free graphs. Lemma 2.1.
Every theta and every prism contains an even hole. proof —
It follows by the fact that there exist two paths in a theta or ina prism that have same parity, which induce an even hole. (cid:3) Even-hole-free graphs excluding a minor
In this section we prove an ‘induced grid theorem’ for graphs excluding afixed minor. From this we derive that even-hole-free graphs excluding afixed minor have bounded tree-width.We begin by defining grids and walls. Let n, m be integers with n, m ≥ n × m ) -grid is the graph G n × m with V ( G n × m ) = [ n ] × [ m ] and E ( G n × m ) = (cid:8) { ( i , j ) , ( i , j ) } | | i − i | + | j − j | = 1 , i , i ∈ [ n ] , j , j ∈ [ m ] (cid:9) . Figure 2 shows G × .Figure 2: The (5 × G × .Let n, m ≥ elementary ( n × m ) -wall is a graph G =( V, E ) with vertex set V = (cid:8) (1 , j − | ≤ j ≤ m (cid:9) ∪ (cid:8) ( i, j ) | < i < n, ≤ j ≤ m (cid:9) ∪ (cid:8) ( n, j − | ≤ j ≤ m, if n is even (cid:9) ∪ (cid:8) ( n, j ) | ≤ j ≤ m, if n is odd (cid:9) and edge set E = (cid:8) (1 , j − , (1 , j + 1) | ≤ j ≤ m − (cid:9) ∪ (cid:8) { ( i, j ) , ( i, j + 1) } | ≤ i < n, ≤ j < m (cid:9) ∪ (cid:8) { ( n, j ) , ( n, j + 2)) } | ≤ j < m if n is odd (cid:9) ∪ (cid:8) { ( n, j − , ( n, j + 1) } | ≤ j < m if n is even (cid:9) ∪ (cid:8) { ( i, j ) , ( i + 1 , j ) } | ≤ i < n, ≤ j ≤ m, i, j odd (cid:9) ∪ (cid:8) { ( i, j ) , ( i + 1 , j ) } | ≤ i < n, ≤ j ≤ m, i, j even (cid:9) . Figure 3 shows an elementary (5 × n × m )-wall has n horizontal paths, where the first hor-izontal path is induced by the vertex set (cid:8) (1 , j − | ≤ j ≤ m (cid:9) , the i th horizontal path is induced by the vertex set (cid:8) ( i, j ) | ≤ j ≤ m (cid:9) .for 1 < i < n , and the n th horizontal path is induced by the vertex set (cid:8) ( n, j − | ≤ j ≤ m (cid:9) if n is odd, and by (cid:8) ( n, j ) | ≤ j ≤ m (cid:9) if n is even. An elementary ( n × m )-wall has m vertical paths, where the j thvertical path is induced by the vertex set (cid:8) ( i, j − , ( i + 1 , j − | ≤ i 1) and ( n, m ) are called the corners of the (triangulated) grid.The following lemma shows that a large wall contains many smallerinduced subwalls. Lemma 3.2. For n, m ∈ N , let W be an ( n × m ) -wall. Let X be a setof pairwise non-adjacent rows of W and let Y be a set of pairwise non- × G (cid:52) × . adjacent columns of W . Then W contains an induced ( | X | × | Y | ) -wall W (cid:48) ,with V ( W (cid:48) ) ⊆ (cid:83) P ∈ X V ( P ) ∪ (cid:83) Q ∈ X V ( Q ) . proof — Assume that | X | ≥ | Y | ≥ 2. We obtain W (cid:48) by taking thesubgraph of W induced by the set (cid:83) P ∈ X V ( P ) ∪ (cid:83) Q ∈ X V ( Q ) and repeatedlydeleting degree-1-vertices until all vertices have degree at least 2. (cid:3) For the proof of Theorem 1.1, we use a corollary of the main result of[11], and we need the notion of contraction . Let G and H be graphs. If H can be obtained from G by a sequence of edge contractions, then H is calleda contraction of G . Alternatively, contractions can be defined via mappingsas follows. Let G and H be graphs and let ϕ : V ( G ) → V ( H ) be a surjectivemapping such that1. for every vertex v ∈ V ( H ), its pre-image ϕ − ( v ) is connected in G ,2. for every edge uv ∈ E ( H ), the graph G [ ϕ − ( u ) ∪ ϕ − ( v )] is connected,3. for every edge xy ∈ E ( G ), either ϕ ( u ) = ϕ ( v ) or ϕ ( u ) ϕ ( v ) ∈ E ( H ). Corollary 3.3 (Fomin, Golovach and Thilikos [11]) . Let H be a graph andlet G be a graph excluding H as a minor. There exists a constant c H suchthat if tw( G ) ≥ c H · ( k + 1) , then G contains an induced subgraph thatcontains G (cid:52) k × k as a contraction. The following lemma will help us to find a large stone wall in a graphcontaining a large triangulated grid as a contraction. Lemma 3.4. Let G be a connected graph whose vertex set is partitionedinto connected sets A , A (cid:48) , B , B (cid:48) , C , C (cid:48) and S . Suppose that every edge of G has either both ends in one of the sets, or is from A (cid:48) to A , from B (cid:48) to B ,from C (cid:48) to C , or from S to A ∪ B ∪ C .If a ∈ A (cid:48) , b ∈ B (cid:48) and c ∈ C (cid:48) , then a , b and c are the degree one verticesof some induced fork or semi-fork of G . proof — Let P be a shortest path from b to c in G [ B (cid:48) ∪ B ∪ S ∪ C ∪ C (cid:48) ].Note that P must go through S . Let Q = a . . . w in G [ A (cid:48) ∪ A ∪ S ] be ashortest path such that w has some neighbors in P . Let u (resp. v ) be the11eighbor of w in P closest to b (resp. to c ) along P . Note that u (cid:54) = b and v (cid:54) = c . If u = v , then P and Q form a fork. If u is adjacent to v , then P and Q form a semi-fork. If u (cid:54) = v and uv / ∈ E ( G ), then aP w , bQu and cQv form a fork. In all cases, a , b and c are the three vertices of degree 1 of thefork or semi-fork. (cid:3) For tidying up stone walls, we make use of a natural variant of Ramsey’sTheorem for bipartite graphs, first introduced by Beineke and Schwenk in1975. Theorem 3.5 (Beineke and Schwenk [2]) . For every integer r ≥ thereexists a smallest positive integer n = n ( r ) , such that any 2-edge-coloring ofthe complete bipartite graph K n,n contains a monochromatic K r,r . In [23] it was shown that n ( r ) ≤ r ( r − 1) + 1.The next lemma shows that any sufficiently large stone wall also containsa large homogeneous stone wall as induced subgraph. Lemma 3.6. For every integer r ≥ there exists an integer n = n ( r ) suchthat every ( n × n ) -stone wall contains a homogeneous ( r × r ) -stone wall asinduced subgraph. proof — Given r , let n = n ( r ) be large enough. Given an ( n × n )-stone wall W ,we define an auxiliary wall W (cid:48) , which is obtained from W by contractingevery triangle. Each vertex in W (cid:48) that is the result of contracting a triangleis colored red (red encodes ‘semi-fork’), and all other degree-3-vertices of W (cid:48) are colored green (green encodes ‘fork’).Define a complete bipartite graph H with V ( H ) = A ∪ B as follows.The elements of A are horizontal paths of W (cid:48) , and the elements of B are thevertical paths in W (cid:48) . Note that each vertical path has two colored verticesin common with each horizontal path.We fix an orientation of the horizontal paths ‘from left to right’. Nowwe color the edges of H with four colors. Let P ∈ A be a horizontal pathand let Q ∈ B be a vertical path.1. If V ( P ) ∩ V ( Q ) ⊆ V ( W (cid:48) ) contains two green vertices, we color the edge P Q green.2. If V ( P ) ∩ V ( Q ) ⊆ V ( W (cid:48) ) contains two red vertices, we color the edge P Q red.3. If V ( P ) ∩ V ( Q ) ⊆ V ( W (cid:48) ) contains a green and a red vertex, and thegreen vertex appears before the red vertex when traversing P from left toright, then we color the edge P Q white.4. If V ( P ) ∩ V ( Q ) ⊆ V ( W (cid:48) ) contains a green and a red vertex, and the redvertex appears before the green vertex when traversing P from left to right,then we color the edge P Q black. 12y applying Theorem 3.5 (multiple times, if necessary), we find that H contains a large monochromatic complete bipartite subgraph H (cid:48) .If H (cid:48) is green (or red, respectively), we find a large subwall in W (cid:48) whereall vertices of degree 3 are green (red, respectively) as follows. We takethe horizontal and vertical paths in W (cid:48) that correspond to V ( H (cid:48) ), leavingout every second path to make sure that the horizontal paths we keep arepairwise non-adjacent, and that the vertical paths we keep are pairwise non-adjacent. Then we apply Lemma 3.2. Undoing the contractions of trianglesin the case that H (cid:48) is red, we thus obtain a large induced homogeneous stonewall in W .In the case that H (cid:48) is white or black, we find a large subwall W (cid:48)(cid:48) in W (cid:48) where both red and green appear at each intersection of a horizontal anda vertical path. W. l. o. g. assume that H (cid:48) is white (otherwise flip the wallexchanging left and right). We will now explain how to find a large inducedsubwall of W .Let X be a maximal subset of horizontal paths of W (cid:48)(cid:48) of pairwise distance10, and let Y be a maximal subset of vertical paths of W (cid:48)(cid:48) of pairwisedistance 10. Whenever a path P ∈ X and a path Q ∈ Y intersect, wereroute the two paths locally around their intersection to avoid red verticesof degree 3 as follows.Let u , v , w , x be consecutive degree-3-vertices on Q with u, v on P .Assume u , v , w , x appear in this order when walking along Q from top tobottom, and w l. o. g. assume u is red (otherwise walk along Q from bottomto top). Since H (cid:48) is white, w is green x is red. Now we reroute P and Q locally, such that after rerouting, both degree-3-vertices at the intersectionof P and Q are green. The rerouting is shown in Figure 7. Note that there isenough space around the intersection, because we only use paths in X ∪ Y . uvw x uvw x Figure 7: Rerouting in the proof of Lemma 3.6.Rerouting in this manner for every pair of paths in X and Y , we end upwith a large subwall of W (cid:48) that is green, which is also an induced homoge-neous stone wall in W . (cid:3) proof — Proof of Theorem 1.1 Let H be a graph and let G be a graphexcluding H as a minor. We may assume G is connected. Let c H be as inCorollary 3.3 and let h ∈ N be sufficiently large, and let k = 8 h . Assumetw( G ) ≥ c H · ( k +1) . Then G contains an induced subgraph G (cid:48) , such that G (cid:48) buc vwAA (cid:48) SC B (cid:48) C (cid:48) BPQ Figure 8: Proof of Theorem 1.1: Using Lemma 3.4 to find an induced forkor semi-fork in G (cid:48) .contains G (cid:52) k × k as a contraction, witnessed by a contraction mapping ϕ : G (cid:48) → G (cid:52) k × k . The graph G (cid:52) k × k contains (2 h ) graphs G (cid:52) × . We pick every secondrow of graphs G (cid:52) × , and every second graph G (cid:52) × of the row allows us to findan induced fork or an induced semi-fork in G (cid:48) as follows. Assume the verticesof G (cid:52) × are (1 , , . . . , (4 , A (cid:48) := ϕ − ((1 , , A := ϕ − ((2 , , B (cid:48) := ϕ − ((1 , , B := ϕ − ((1 , , C (cid:48) := ϕ − ((4 , , C := ϕ − ((3 , , and S := ϕ − ((2 , G (cid:48) [ A ∪ A (cid:48) ∪ B ∪ B (cid:48) ∪ C ∪ C (cid:48) ∪ S ] and hence in G (cid:48) (cf. Figure 8).These forks can be combined into a large stone wall by adding inducedpaths to connect the forks or semi-forks appropriately, cf. Figure 9.Hence G contains a large stone wall as induced subgraph, and we canuse Lemma 3.6 to complete the proof. (cid:3) Let us remark that the function f H in Theorem 1.1 is computable. Corollary 3.7. For every fixed graph H , the class of (theta, prism)-freegraphs that do not contain H as a minor has bounded tree-width. In partic-ular, even-hole-free graphs that do not contain H as a minor have boundedtree-width. proof — A large wall contains a theta, and the line graph of a large wallcontains a prism. (cid:3) The following corollary reproves a theorem from [21].14igure 9: Proof of Theorem 1.1: Finding a stone wall in G (cid:52) k × k . Corollary 3.8. Planar even-hole-free graphs have bounded tree-width. proof — This follows from Corollary 3.7 because planar graphs exclude K as a minor. (cid:3) In this Section, we prove that even-hole subcubic graphs can be describedby a structure theorem, that implies tree-width at most 3. In fact our resultis for a more general class: (theta, prism)-free subcubic graphs.A wheel that is not a pyramid is a proper wheel . A sector of a wheel( H, x ) is a subpath of H whose endnodes are adjacent to x , and whoseinternal vertices are not.An extended prism is a graph made of five vertex-disjoint chordless pathsof length at least 1 A = a . . . x , A (cid:48) = x . . . a (cid:48) , B = b . . . y , B (cid:48) = y . . . b (cid:48) , C = c . . . c (cid:48) such that abc is a triangle, a (cid:48) b (cid:48) c (cid:48) is a triangle, xy is an edge andno edges exist between the paths except xy and those of the two triangles(see Figure 10).A subset (possibly empty) of vertices S ⊆ V ( G ) is a separator of G if G \ S contains at least two connected components. A clique separator is aseparator S that is a clique. 15 a ac (cid:48) yb (cid:48) bcy b (cid:48) bc c (cid:48) a (cid:48) xa H Figure 10: Two different drawings of an extended prismA proper separation in a graph G is a triple ( { a, b } , X, Y ) satisfying thefollowing.(i) { a, b } , X , Y are disjoint, non-empty and V ( G ) = { a, b } ∪ X ∪ Y .(ii) There are no edges from X to Y .(iii) a and b are non-adjacent.(iv) a and b have exactly two neighbors in X .(v) a and b have exactly one neighbor in Y .(vi) There exists a path from a to b with interior in X , and there exists apath from a to b with interior in Y .(vii) G [ Y ∪ { a, b } ] is not a chordless path from a to b .A proper separator of G is a pair { a, b } ⊆ V ( G ) such that there exists aproper separation ( { a, b } , X, Y ).Let C be the class of (theta, prism)-free subcubic graphs. The cube isthe graph made of a hole v v . . . v v and two non-adjacent vertices x and y such that N H ( x ) = { v , v , v } and N H ( y ) = { v , v , v } . Call a graphin C basic if it is isomorphic to a chordless cycle, a clique of size at most 4,the cube, a proper wheel, a pyramid, or an extended prism. An example ofgraph in C that is not basic is provided in Figure 11.We need the following lemma. Lemma 4.1. Let G be a theta-free subcubic graph, let H be a hole in G , and v ∈ G \ H . Then v has at most three neighbors in H , and if v has exactlytwo neighbors in H , then they are adjacent. proof — Let v ∈ G \ H . Since G is subcubic, d H ( v ) ≤ 3. If v has exactlytwo neighbors in H , but they are non-adjacent then G [ H ∪{ v } ] would inducea theta, a contradiction. (cid:3) The main theorem of this section is the following.16 x x y x n − p (cid:48) b (cid:48) p (cid:48) a (cid:48) p (cid:48) a (cid:48) p (cid:48) b (cid:48) p (cid:48) b (cid:48) = p (cid:48) a (cid:48) + 1 p (cid:48) a (cid:48) p (cid:48) a (cid:48) n − p (cid:48) a (cid:48) n − +1 p (cid:48) b (cid:48) n − − p (cid:48) b (cid:48) n − p a p a p b p b n p a n R (cid:48) Q (cid:48) Q n − R n − Q Q (cid:48) n p (cid:48) a (cid:48) n = p (cid:48) b (cid:48) n x n uvp a = p b p a +1 p b − p a n − p b n − = p a n − +1 y n − Figure 11: An example of non-basic graph in C Theorem 4.2. Let G be a (theta, prism)-free subcubic graph. Then one ofthe following holds: • G is a basic graph; • G has a clique separator of size at most 2; • G has a proper separator. proof — Let G be a (theta, prism)-free subcubic graph. We may assumethat G has no clique separator (and is in particular connected for otherwisethe empty set is a clique separator).(1) We may assume that G is ( K , cube)-free.Proof of (1). If G contains K , then since G is a subcubic connected graph, G = K , so G is basic. The proof is similar when G contains the cube. Thisproves (1).(2) We may assume that G does not contain a proper wheel.Proof of (2). Let W = ( H, x ) be a proper wheel in G . Let a, b, c , be thethree neighbors of x . We call A ( B , C , resp.) the path of H from b to c (from a to c , from a to b , resp.) that does not contain a ( b , c , resp.).Suppose that some vertex y of G \ W has neighbors in the three sectorsof W , say a (cid:48) in A , b (cid:48) in B , and c (cid:48) in C . Hence, a , c (cid:48) , b , a (cid:48) , c , and b (cid:48) appear inthis order along H . If ac (cid:48) / ∈ E ( G ), then xaBb (cid:48) , xcBb (cid:48) , and xbCc (cid:48) yb (cid:48) inducea theta, so ac (cid:48) ∈ E ( G ). Symmetrically, c (cid:48) b , ba (cid:48) , a (cid:48) c , b (cid:48) c , and b (cid:48) a are all17n E ( G ), so H , x , and y induce a cube, a contradiction to (1). It followsthat every vertex has neighbors in at most two sectors of W .If G = W , then G is basic, so let L be a component of G \ W . Notethat N ( L ) contains at least two vertices since G has no clique separator.If N ( L ) is included in a sector of W , then the ends of this sector form aproper separator. We may therefore assume that N ( L ) intersects at leasttwo sectors of W .Since L is connected, it contains a path P = u . . . v such that u hasneighbors in a sector of W (say C up to symmetry), and v has neighbors inanother sector of W (say A up to symmetry). Suppose that P is minimalwith respect to this property. Then either u = v and by the second para-graph of this proof, u has no neighbor in B ; or u (cid:54) = v and, by minimality of P , u has neighbors only in C , v has neighbors only in A , and the interior of P is anticomplete to W . In each case, we let u (cid:48) be the neighbor of u in C closest to a along C and we let v (cid:48) be the neighbor of v in A closest to c along A . Note that because u (cid:48) and v (cid:48) exist, ab / ∈ E ( G ) and bc / ∈ E ( G ). So, ac / ∈ E ( G ) for otherwise, ( W, x ) would form a pyramid and be a non-properwheel. Now, the three paths axc , B , and aCu (cid:48) uP vv (cid:48) Ac form a theta, acontradiction. This proves (2).(3) We may assume that G does not contain an extended prism.Proof of (3). Let W be an extended prism in G , with notation as in thedefinition. Suppose that some vertex z of G \ W has neighbors in threedistinct paths among A, A (cid:48) , B, B (cid:48) , and C , and call Q, R, S these three paths(so { Q, R, S } ⊆ { A, A (cid:48) , B, B (cid:48) , C } ). It is easy to check that some hole H of W contains Q and R . By Lemma 4.1, z must have three neighbors in H ,so H and z form a proper wheel, a contradiction to (2).If G = W , then G is basic, so let L be a component of G \ W . Notethat N ( L ) contains at least two vertices since G has no clique separator. If N ( L ) is included in one of V ( A ), V ( A (cid:48) ), V ( B ), V ( B (cid:48) ), or V ( C ), then theends of this path form a proper separator. We may therefore assume that N ( L ) intersects at least two paths in { A, A (cid:48) , B, B (cid:48) , C } .Since L is connected, it contains a path P = u . . . v such that u hasneighbors in a path Q ∈ { A, A (cid:48) , B, B (cid:48) , C } and v has neighbors in anotherpath R ∈ { A, A (cid:48) , B, B (cid:48) , C } . Suppose that P is minimal with respect tothis property. So by the minimality of P , either u = v and by the secondparagraph of this proof, u = v has no neighbor in { A, A (cid:48) , B, B (cid:48) , C } \ { Q, R } ;or u (cid:54) = v and u has neighbors only in Q , v has neighbor only in R and theinterior of P is anticomplete to W .Note that each of N Q ( u ) and N R ( v ) is a vertex or an edge. For otherwise,suppose that u has two non-adjacent neighbors in Q (resp. in R ). Since G issubcubic and Q (resp. R ) can be completed to a hole J of W , by Lemma 4.1, u has three pairwise non-adjacent neighbors in J , so G contains a properwheel, a contradiction to (2). We may now break into four cases.18ase 1: { Q, R } = { A, A (cid:48) } or { Q, R } = { B, B (cid:48) } . Up to symmetry, wesuppose Q = A and R = A (cid:48) . Then, P can be used to find a path from a to a (cid:48) that does not contain x , and that together with B , B (cid:48) and C form aprism, a contradiction.Case 2: { Q, R } = { A, B } or { Q, R } = { A (cid:48) , B (cid:48) } . Up to symmetry, wesuppose Q = A and R = B . If u has two adjacent neighbors in A , then A , A (cid:48) , C , a subpath of B , and P form a prism. So, u has exactly one neighborin A , and symmetrically, v has exactly one neighbor in B . So, A , B , and P form a theta.Case 3: { Q, R } = { A, B (cid:48) } or { Q, R } = { B, A (cid:48) } . Up to symmetry, wesuppose Q = A and R = B (cid:48) . If u has two adjacent neighbors in A , then A , A (cid:48) , C , a subpath of B (cid:48) , and P form a prism. So, u has exactly one neighborin A , and symmetrically, v has exactly one neighbor in B (cid:48) . So, A , B (cid:48) , C ,and P form a theta.Case 4: { Q, R } is one of { A, C } , { A (cid:48) , C } , { B, C } or { B (cid:48) , C } . Up to symme-try, we suppose Q = A and R = C . If v has two adjacent neighbors in C ,then C , B , B (cid:48) , a subpath of A and P form a prism. So, v has exactly oneneighbor in C . So, C , B , A (cid:48) , a subpath of A , and P form a theta.This proves (3).(4) We may assume that G does not contain a pyramid.Proof of (4). Let W be a pyramid with notation as in the definition. Firstnote that a vertex v ∈ V ( G \ W ) cannot have neighbors in the three paths P , P , and P , for otherwise there exists a theta from v to x .If G = W , then G is basic, so let L be a component of G \ W . Notethat N ( L ) contains at least two vertices since G has no clique separator. If N ( L ) is included in one of P , P , or P , then the ends of this path forma proper separator, or a clique separator when this path has length 1. Wemay therefore assume that N ( L ) intersects at least two paths.So, since L is connected, it contains a path P = u . . . v such that u hasneighbors in a path P i (say P up to symmetry), and v has neighbors inanother path P j (say P up to symmetry). Suppose that P is minimal withrespect to this property. So by minimality, either u = v and by the firstparagraph of this proof, u = v has no neighbor in P ; or u (cid:54) = v and u hasneighbors only in P , v has neighbor only in P , and the interior of P isanticomplete to W .Note that each of N P ( u ) and N P ( v ) is a vertex or an edge. If u = v ,this is because G contains no proper wheel by (2). If u (cid:54) = v , this is because u and v have degree at most 3 and we apply Lemma 4.1.If N P ( u ) and N P ( v ) are both edges, then u (cid:54) = v (because G is subcubic),so P , P , and P form a prism. If each of N P ( u ) and N P ( v ) is a vertex,then P , P , and P form a theta. So, up to symmetry, N P ( u ) is a vertex19 (cid:48) , N P ( v ) is an edge yz (where x, y, z, a appear in this order along P ).If u (cid:48) (cid:54) = x , then V ( P ) ∪ V ( W ) \ V ( zP a ) induces a theta from u (cid:48) to x , so u (cid:48) = x . Hence, W and P form an extended prism, a contradiction to (3)This proves (4).(5) We may assume that G does not contain a hole.Proof of (5). Let W be a hole in G . First note that a vertex v ∈ V ( G \ W )cannot have three neighbors in W , for otherwise v and W would form aproper wheel or a pyramid. So, by Lemma 4.1, every vertex of G \ W has atmost one neighbor in W , or exactly two neighbors in W that are adjacent.If G = W , then G is basic, so suppose that L is a component of G \ W .If N ( L ) is included in some edge of W , then G has a clique separator, sosuppose that there exist a, b ∈ V ( W ) that are non-adjacent and that bothhave neighbors in L . Since L is connected, there exists a path P = u . . . v ,such that u is adjacent to a and v is adjacent to b . We suppose that a, b, u, v and P are chosen subject to the minimality of P . Note that u (cid:54) = v since avertex in G \ W cannot have two non-adjacent neighbors in W .Suppose that some internal vertex of P has a neighbor x in W . So x must be adjacent to a , for otherwise a subpath of P from u to a neighbor of x in P contradicts the minimality of P . Similarly, x is adjacent to b . If a and b have two common neighbors in H , say x and y (so W = axbya ), and x and y both have neighbors in the interior of P , then the vertices x and y togetherwith a subpath of P contradict the minimality of P . Hence, x is the uniquevertex of W with neighbors in the interior of P . If u and v have exactlytwo adjacent neighbors in W , then W and P form an extended prism, acontradiction to (3). If exactly one of u or v has exactly two neighborsin W , then W and a subpath of P form a pyramid, a contradiction to (4).So, u and v both have a unique neighbor in W . Now, P and H form aproper wheel, a contradiction to (2).So, the interior of P is anticomplete to W . Hence, P and W form atheta, a prism or a pyramid, in every case a contradiction to G ∈ C , orto (4). This proves (5).(6) We may assume that G does not contain a triangle.Proof of (6). Let W = abc be a triangle in G . If G = W , then G is basic,so suppose that L is a component of G \ W . If | N ( L ) | ≤ 2, then G has aclique separator of size at most 2, so suppose that N ( L ) = { a, b, c } .Let P = u . . . v be a path in L such that u is adjacent to a , v is adjacentto b , and suppose P is minimal. If u (cid:54) = v , then P , a , and b form a hole,a contradiction to (5), so u = v . By (1), u is non-adjacent to c . Hence, apath in L from u to a neighbor of c , together with a , would form a hole, acontradiction to (5). This proves (6).Now, by (5) and (6), G has no cycle. So, G is a tree. It is thereforea complete graph on at most two vertices (that is basic) or it a has clique20igure 12: Two chordal graphs with clique number 4separator of size 1. (cid:3) Let us point out that Theorem 4.2 is a full structural description of theclass of subcubic (theta, prism)-free graphs, in the sense that every graphin the class can be obtained from basic graphs by repeatedly applying someoperations: gluing along a (possibly empty) clique, and an operation calledproper gluing that we describe now.Consider two graphs G and G . Suppose that G contains two non-adjacent vertices a and b of degree 3, and such that a path P from a to b with internal vertices all of degree 2 exists in G . Suppose that G contains two non-adjacent vertices a and b of degree 2, and such that apath P from a to b with internal vertices all of degree 2 exists in G . Let G be the graph obtained from the disjoint union of G and G by removing theinternal vertices of P and P , by identifying a and a , and by identifying b and b . We say that G is obtained from G and G by a proper gluing .We omit the details of the proof and just sketch it. We apply Theo-rem 4.2. If G is basic, there is nothing to prove. If G has a clique separator,it is obtained by two smaller graphs by gluing along a clique. If G hasa proper separation, then it is obtained from smaller graphs by a propergluing. Corollary 4.3. Every subcubic (theta, prism)-free graph (and therefore ev-ery even-hole-free subcubic graph) has tree-width at most 3. proof — The proof is by induction. Let us first prove that all basic graphshave tree-width at most 3. First observe that contracting an edge withone vertex of degree 2 preserves the tree-width. It follows that all basicgraphs, except the cube and the extended prisms, have tree-width at mostthe tree-width of K , that is 3. In Figure 12, we show a chordal graph J with ω ( J ) = 4 that contains the cube or the smallest extended prism as asubgraph, showing that here again the tree-width is at most 3.Also, it is easy to check that the two operations gluing along a cliqueand proper gluing do not increase the tree-width (this can be proved byobserving that the operation are particular cases of what is called clique-21um in the theory of tree-width, or by a direct proof using the definition oftree-width given at the beginning of the paper). (cid:3) Note also that all graphs in C can be proved to be planar by an easyinduction. Our goal in this section is to prove that (even hole, pyramid)-free graphswith maximum degree at most 4 have bounded tree-width. We rely on twoknown theorems that we now explain.Let H be a hole in a graph and let u be a vertex not in H . We say that u is major w.r.t. H if N H ( u ) is not included in a 3-vertex path of H . Weomit “w.r.t. H ” when H is clear from the context. Lemma 5.1. If G be an (even hole, pyramid)-free graph with maximumdegree at most 4, H is a hole of G and v is a vertex that is major w.r.t. G ,then v has exactly three neighbors that are pairwise non-adjacent. proof — Since v is major, it has at least two neighbors in H . If v hasexactly two neighbors in H , since v is major these two neighbors are non-adjacent. Therefore, H and v form a theta, a contradiction. If G has exactlythree neighbors in H , then they are pairwise non-adjacent because v is majorand G has no pyramid. If v has 4 neighbors in H , then H and v form aneven wheel, a contradiction. (cid:3) When H is a hole in some graph and u is a vertex not in H with at leasttwo neighbors in H , we call u -sector of H any path of H of length at least 1,whose ends are adjacent to u and whose internal vertices are not. Observethat H is edgewise partitioned into its u -sectors.Note that by Lemma 5.1, when v is major w.r.t. H , ( H, v ) is a wheel, sothe notion of v -sector in H is equivalent to the notion of a sector the wheel( H, v ). The following appeared in [8]. Theorem 5.2. Let G be a graph with no even hole and no pyramid, H ahole in G and v a major vertex w.r.t. H . If C is a connected componentof G \ N [ v ] , then there exists a v -sector P = x . . . y of H such that N ( C ) ⊆{ x, y } ∪ ( N ( v ) \ V ( H )) . A graph G is a ring if its vertex-set can be partitioned into k ≥ X , . . . , X k such that (the subscript are taken modulo k ):1. X , . . . , X k are cliques;2. for all i ∈ { , . . . , k } , X i is anticomplete to V ( G ) \ ( X i − ∪ X i ∪ X i +1 );22. for all i ∈ { , . . . , k } , some vertex of X i is complete to X i − ∪ X i +1 ;4. for all i ∈ { , . . . , k } and all x, x (cid:48) ∈ X i , either N [ x ] ⊆ N [ x (cid:48) ] or N [ x (cid:48) ] ⊆ N [ x ].A graph G is a if its vertex-set can be partitioned into7 sets X , . . . , X such that (the subscript are taken modulo k ):1. X , . . . , X k are cliques;2. for all i ∈ { , . . . , k } , X i is complete to V ( G ) \ ( X i − ∪ X i ∪ X i +1 );3. for all i ∈ { , . . . , k } , X i is anticomplete to X i − ∪ X i +1 .The following is a rephrasing of Theorem 1.8 in [5]. Note that in [5], thedefinition of rings is slightly more restricted (at least 4 sets are required).We need rings with 3 sets for later use in inductions, and slightly extendingthe notion of ring cannot turn Theorem 1.8 in [5] into a false statement. Theorem 5.3. If G is (theta, prism, pyramid)-free and for every hole H of G , no vertex of G is major w.r.t. H , then G is a complete graph, or G is aring, or G is a 7-hyperantihole, or G has a clique separator. Lemma 5.4. A complete graph, a ring, or a 7-antihole of maximum degreeat most 4 does not contain K as a minor. proof — For complete graphs, this is obvious since K is the biggest com-plete graph of maximum degree at most 4. For 7-hyperantiholes, the proofis also easy because each of the cliques in the definition must be on a singlevertex, so that | V ( G ) | = 7, and a K minor obviously does not exists.So, suppose that G is a ring of maximum degree at most 4 (we use for G the notation as in the definition of rings).(1) For all i ∈ { , . . . , k } , one of X i − , X i or X i +1 contains only one vertex.Moreover, if | X i | = 3 , then X i − , X i +1 and X i +2 all contain only one vertex.Proof of (1). Otherwise, some vertex in X i or X i +1 has degree 5, a contra-diction to our assumption. This proves (1).We now prove by induction on k (the number of sets in the ring) that G does not contain K as a minor. If k equals 3 or 4, then by (1), we seethat | V ( G ) | ≤ 6, so G does not contain K as a minor since G is not K . If k ≥ 5, then by (1), there exist two distinct sets of the ring X i , X j that areanticomplete to each other and such that X i = { x } and | X j | = { x (cid:48) } . Bythe definition of rings, G \ { x, x (cid:48) } has two connected components C and C (cid:48) ,and it is straightforward to check the two graphs G C and G C (cid:48) obtained from G [ C ∪ { x, x (cid:48) } ] and G [ C (cid:48) ∪ { x, x (cid:48) } ] respectively by adding an edge between x and x (cid:48) are rings (this is the place where we need a ring on three sets). Also,23t is straightforward to check that a K minor in G yields a K minor in oneof G C or G C (cid:48) , a contradiction to the induction hypothesis. (cid:3) We can now prove the main theorem of this section. Theorem 5.5. If a graph G is (even hole, pyramid)-free with maximumdegree 4, then G contains no K as a minor. proof — Suppose that G is a counter-example with a minimum numberof vertices. So, G contains K as a minor. By the minimality of G and thedefinition of minors, it follows that V ( G ) can be partitioned into six non-empty sets B , . . . , B such that for all i, j ∈ { , . . . , } , G [ B i ] is connectedand there is at least one edge between B i and B j . Case 1: G contains no hole with a major vertex.By Theorem 5.3 and Lemma 5.4, G has a clique separator K . It isstraightforward to check that for one component C of G \ K , the graph G [ K ∪ C ] contains K as a minor, a contradiction to the minimality of G . Case 2: G contains a hole H and a vertex v that is major w.r.t. H .By Lemma 5.1, v has exactly three neighbors in H that are pairwisenon-adjacent. Possibly, v has a neighbor w / ∈ H (if v has degree three, weset v = w ). Let a, b, c be the three neighbors of v in H .Up to symmetry, B and B do not contain a, b, c, w . So, some connectedcomponent C of G \ { v, w, a, b, c } contains B ∪ B . By Theorem 5.2, thereexists a v -sector, say P = a . . . b up to symmetry, of H such that N ( C ) ⊆{ a, b, w } . Note that if v = w , then { a, b } is a separator of G , in which casethe proof is easier. So in what follow, a reader may assume for simplicitythat v (cid:54) = w , though what is written is correct even if v = w .Let C (cid:48) be the union of all components X of G \ { v, w, a, b, c } such that N ( X ) ⊆ { a, b, w } . Let D be V ( G ) \ ( C (cid:48) ∪ { a, b, w } ). Note that C ⊆ C (cid:48) and v, c ∈ D . Note that B ∪ B ⊆ C (cid:48) , and since { a, b, w } separates C (cid:48) from D ,we may assume that B ⊆ C (cid:48) .Let S a (resp. S b ) be the v -sector of H from a (resp. b ) to c . Let G (cid:48) bethe graph obtained from G [ C (cid:48) ∪ { a, b, c, v, w } ] by adding the edges ca and cb . Also, the edge cw is added to G (cid:48) if and only if w has a neighbor in theinterior of the path formed by S a and S b .(1) G (cid:48) is (even hole, pyramid)-free and has maximum degree at most 4.Proof of (1). Clearly G has maximum degree at most 4. Since G (cid:48) \ c is aninduced subgraph of G , every even hole or pyramid of G (cid:48) goes through c .Suppose that J is an even hole of G (cid:48) . Since it goes through c , up tosymmetry, we may assume that J goes through cb . If J contains a , then G is formed of a , c , b and a path P of even length from a to b . So, P , S a and S b form an even hole of G , unless w ∈ V ( J ) and w has a neighbor in the interiorof the path induced by S a ∪ S b . But this case leads to a contradiction, since24y the definition of G (cid:48) , we would have cw ∈ E ( G (cid:48) ), so J would not be a holeof G (cid:48) .So, J does not go through a . It follows that J is formed by bc and apath Q of odd length from w to b . Note that in this case, cw ∈ E ( G ) and v / ∈ V ( J ), so v (cid:54) = w . It follows that Q and v form an even hole of G (cid:48) , acontradiction.Suppose that G (cid:48) contains a pyramid Π. Since Π contains c , it doesnot contain v because v dominates c , in G (cid:48) , ie N G (cid:48) [ c ] ⊆ N G (cid:48) [ v ] (and in apyramid, no vertex dominates another vertex). If we replace c by v in Π,then we obtain an induced subgraph of G that is not a pyramid since G ispyramid-free. This implies that cw / ∈ E ( G (cid:48) ). So, c has degree 2 in Π and w has no neighbor in the interior of the path induced by S a ∪ S b . Hence,replacing acb by S a and S b in Π yields a pyramid of G , a contradiction. Thisproves (1).(2) G (cid:48) contains K as a minor.Proof of (2). Suppose that a ∈ B , w ∈ B and b ∈ B . We then set B (cid:48) = ( B \ D ) ∪ { c } , B (cid:48) = ( B \ D ) ∪ { v } and B (cid:48) = B \ D . We observe thatthere are edges from B (cid:48) to B (cid:48) , from B (cid:48) to B (cid:48) and from B (cid:48) to B (cid:48) . Also,each of these sets is connected in G (cid:48) , and together with B , B and B theyform a K minor of G (cid:48) . We may therefore assume that B ∩ { a, b, w } = ∅ ,so that B ⊆ C (cid:48) .If { a, b, w } ∩ B = { a, b } , then { a, b, w } ∩ B = { w } . We then set B (cid:48) = ( B \ D ) ∪ { c } and B (cid:48) = ( B \ D ) ∪ { v } . We observe that there areedges from B (cid:48) to B (cid:48) . Also, these sets are connected in G (cid:48) , and together with B , B , B and B they form a K minor of G (cid:48) . Hence, we may assume that { a, b, w } ∩ B (cid:54) = { a, b } , and symmetrically { a, b, w } ∩ B (cid:54) = { a, b } .We may assume that a, w ∈ B . We set B (cid:48) = ( B \ D ) ∪ { c, v } and B (cid:48) = B \ D . We observe that there are edges from B (cid:48) to B (cid:48) . Also, thesesets are connected in G (cid:48) , and together with B , B , B and B they form a K minor of G (cid:48) . This proves (2).Since G (cid:48) is smaller than G , (1) and (2) contradict the minimality of G . (cid:3) In the next corollary, we use the function f H ( k ) as defined in Theo-rem 1.1. Corollary 5.6. Every (even hole, pyramid)-free graph of maximum degreeat most 4 has tree-width less than f K (3) . proof — Suppose that G has tree-width at least f K (3). By Theorem 5.5, G does not contain K as a minor. By Theorem 1.1, G contains a (3 × × × × (cid:3) × × In this section, we investigate a possible structure theorem that would de-scribe even-hole-free graphs with maximum degree at most 4. We call pat-terns , the graphs that are represented on Figure 14 and Figure 15. Say thata graph is basic if it is a complete graph or a chordless cycle, or it can beobtained from one of the patterns, by replacing dashed lines with paths oflength at least one or contracting some dashed lines into single vertices. Webelieve that an even-hole-free graph with maximum degree 4 must be eitherbasic or decomposable with a clique separator or a 2-join that we definebelow.A in a graph G is a partition of V ( G ) into two sets X , X eachof size at least 3, such that for i = 1 , X i contains two non-empty disjointsets A i , B i , A is complete to A , B is complete to B , and there are noother edges between X and X . Moreover, for i = 1 , X i does not consistof a path with one end in A i , one end in B i and no internal vertex in A i ∪ B i .We are not sure that our list of patterns is complete for our class, butwe believe that the real list is close to it and, above all, finite. This shouldimply that the tree-width is bounded. Also, we wonder whether a similarapproach can be extended to even-hole-free graphs of maximum degree k for any fixed integer k . Observe that for k = 3, this is what we actually doin Theorem 4.2, since the list of basic graphs can be seen as obtained by afinite list of patterns and the so-called proper separator is a special case of2-join. For k ≥ 5, rings (already defined in Section 5) become a problem,but an extension of the notion of 2-join might lead to a true statement. References [1] Isolde Adler and Frederik Harwath. Property testing for bounded de-gree databases. 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