A more accurate view of the Flat Wall Theorem
AA more accurate view of the Flat Wall Theorem
Ignasi Sau Giannos Stamoulis Dimitrios M. Thilikos Abstract
We introduce a supporting combinatorial framework for the Flat Wall Theorem. In particular,we suggest two variants of the theorem and we introduce a new, more versatile, concept of wallhomogeneity as well as the notion of regularity in flat walls. All proposed concepts and resultsaim at facilitating the use of the irrelevant vertex technique in future algorithmic applications.
Keywords : graph minors; treewidth; Flat Wall Theorem; parameterized algorithms; irrelevantvertex technique; homogeneous walls. LIRMM, Université de Montpellier, CNRS, Montpellier, France. Supported by the ANR projects DEMOGRAPH(ANR-16-CE40-0028), ESIGMA (ANR-17-CE23-0010), ELIT (ANR-20-CE48-0008), the French-German Collabora-tion ANR/DFG Project UTMA (ANR-20-CE92-0027), and the French Ministry of Europe and Foreign Affairs, viathe Franco-Norwegian project PHC AURORA. Emails: [email protected] , [email protected] LIRMM, Université de Montpellier, Montpellier, France. Email: [email protected] a r X i v : . [ c s . D M ] F e b ontents Introduction
One of the cornerstone achievements of the Graph Minors series by Robertson and Seymour wasthe celebrated
Flat Wall Theorem , proved in the 13th paper of the series [36]. It is a powerful graphstructural result, revealing the local structure of H -minor-free graphs. The Flat Wall Theorem hasimportant consequences and applications in structural graph theory and in graph algorithm design.It served as the combinatorial base for the design of an algorithm for the following two problems:• Minor Testing : Given a graph G and a k -vertex graph H, decide whether G contains H as a minor.• Disjoint Paths : Given a graph G with k pairs of terminals ( s i , t i ) , . . . , ( s k , t k ) , decidewhether G contains k vertex-disjoint paths joining s i and t i for every i ∈ { , . . . , k } . These algorithms run in time f ( k ) · n on n -vertex graphs, for some function f : N → N (see [27] forquadratic-time improvements). This, using the terminology of parameterized complexity, impliesthat both above problems, when parameterized by k, belong to the parameterized class FPT or,alternatively, admit
FPT -algorithms . In order to obtain these algorithms, Robertson and Seymourintroduced a powerful technique, called the irrelevant vertex technique , which has now becomea standard technique in the design of parameterized algorithms (see e.g., Section 7 of the text-book [8]). Further algorithmic applications combining the Flat Wall Theorem and the irrelevantvertex technique appeared later in [2, 9, 12, 17, 25], while generalizations to directed graphs haverecently appeared in [13, 20].
The original statement of the Flat Wall Theorem, as appeared in [36], is the following.
Proposition 1.
There exist functions f : N → N and f : N → N such that if G is a graph and h and k are integers, then one of the following holds:1. G contains K h as a minor .2. G has treewidth at most f ( k, h ) . G has a vertex set A with | A | ≤ f ( h ) , such that G \ A contains a flat wall W of height k. We postpone the formal definitions of “treewidth”, the related concept of “tree decomposition”,and “flat wall” to Section 2. One can get a quick idea of a wall by looking at Figure 1 and offlat wall by looking at Figure 3 and Figure 5. Intuitively, a flat wall W is contained in a largergraph, its compass , that is separated from the rest of the graph via a separator S that is a “suitablychosen” part of the perimeter of W. This compass is “flat” in the sense that it does not containtwo disjoint paths whose endpoints are in S and are “crossing” with respect to the cyclic orderinginduced in S by the perimeter of W. As proved by Kawarabayashi, Thomas, and Wollan [31],this flatness property can be certified by a concept called rendition (corresponding to the concept I.e., some subgraph of G can be contracted to a complete graph on h vertices. rural division in [36]) that can be seen as a plane embedding inside a disk of a hypergraphwith hyperedges of arity at most three (see Figure 2 for a visualization of a rendition). Then thecompass is “embedded” inside the rendition so that it can be seen as the union of graphs called flaps bijectively mapped to the hyperedges of the rendition.In its original version in [36], Proposition 1 was proved for f ( h ) = (cid:0) h (cid:1) with the additionalassertion that f ( k, h ) is a bound on the treewidth of the “internal flaps”, i.e., those that are notincident to the perimeter of W . Later, in [14], the same result was proved (without an algorithm) for f ( h ) = h − f ( k, h ) = O h ( k ) . The original result of Robertson and Seymour was accompaniedwith an O ( n · m )-time algorithm that outputs a certifying structure for each possible outcome. Thisalgorithm was further improved to a linear one by Kawarabayashi, Kobayashi, and Reed in [27].A recent wave of improvements of Proposition 1 appeared in the following form [7, 31]. Proposition 2.
There exist functions f : N → N and f : N → N such that if G is a graph and h and k are integers, and G contains a wall W of height f ( k, h ) as a subgraph, then one of thefollowing holds:1. G contains K h as a minor.2. G has a vertex set A with | A | ≤ f ( h ) , such that G \ A contains a flat wall W of height k. Notice that Proposition 2 can indeed be seen as an extension of Proposition 1 because theexclusion of a wall of height k in a K h -minor-free graph implies that its treewidth is bounded by O h ( k ) [10, 26]. Moreover, according to [31, Theorem 1.9], Proposition 2 holds for f ( h ) = O ( h )and f ( k, h ) = O ( h ( h + r )), and it enjoys the following additional features:(A) In the first case, the graph K h is a minor of G in a way that is “grasped by the wall W ’.’ (B) In the second case, the flat wall W is a subwall of W. (C) Proposition 2 comes with an algorithm that certifies one of the two outcomes in linear time,in particular, in O ( h · m + n ) time.Moreover, the same result with Features (A) and (B) is proved in [31, Theorem 1.7] with the optimalfunction f ( h ) = h − f ( k, h ) . Also [31, Theorem 1.8] correspondsto Proposition 2 with the additional feature that the compass of the flat wall W contains no wallof height f ( k, h ) + 1 , again at the cost of a worst bound for f ( k, h ) . Later, Chuzhoy [7] drastically improved the bounds of Proposition 2 with the extra Features (A)and (B) to f ( h ) = h − f ( k, h ) = O ( h · ( h + k )) . Moreover, Chuzhoy gives a polynomial-timealgorithm for her improved variant, however she does not specify whether this algorithm can runin linear time, as the one in [31, Theorem 1.9]. The notation ‘ O h ( · )’ means that the hidden constants depend only on h. In this paper we always denote by n and m the number of vertices and edges, respectively, of the graph underconsideration. We avoid here the formal definition of “grasping by a wall” as we do not make use of it in this paper; see [31]for the details. However, we stress that it provides additional information that is used in further applications (seee.g., [32]). .2 Our contribution In this paper we provide a series of enhanced algorithmic versions of the Flat Wall Theorem as wellas a series of combinatorial tools related to the applicability of the irrelevant vertex technique. Inour presentation we adopt the framework and the terminology of [31]. Our aim is to introduce a“more accurate” view of the Flat Wall Theorem that, we hope, will be useful for future algorithmicapplications. Our contribution consists in the following. ( α ) Subwalls of flat walls are not always flat. Our initial motivation comes from the fact thatthe claimed Feature (B) in Proposition 2, as stated in [31], needs some slight (but not neglectable)revision. This feature is based on [31, Lemma 6.1], asserting that if W is a flat wall and W isa subwall of W that is disjoint from the perimeter of W, then W is also a flat wall of G. As weobserve in Subsection 2.3, there are some very marginal cases where a subwall of a flat wall is notflat anymore. This phenomenon is illustrated in the flat wall of Figure 3 (in Subsection 2.3). ( β ) A reparation framework. Fortunately, the issue raised in ( α ) is just a minor formal mis-match that harms neither the spirit of the proofs of [31] nor the “essential” correctness of subsequentresults that are based on [31]. The first contribution of our paper is to propose an extension ofthe framework of [31] that supports a formally correct statement of Feature (B) in Proposition 2.What we show (Theorem 5) is that if a wall W is a flat wall, whose flatness is certified by somerendition R , and W is a subwall of W, then there is another, slightly different, subwall ˜ W of W, which we call a W -tilt , that is indeed flat . By the term “slightly different” we mean that W and its W -tilt ˜ W may differ only perimetrically. Moreover, the rendition certifying the flatnessof ˜ W maintains all the “internal” structure of the rendition R , relatively to W . This implies thatall the arguments based on Proposition 2 of [31] are essentially correct, and can become formallycorrect under the suggested framework. In our definitions and proofs we pay attention to all thenecessary formalism so to facilitate dealing with future results that may use those of [31] (or [7]).We conclude with Proposition 7 that is a version of Proposition 2 translated into our framework. ( γ ) A Flat Wall Theorem with compasses of bounded treewidth. Our next result, The-orem 8 in Subsection 3.2, is an improved version of Proposition 1 with the following additionalfeatures: (1) f ( k, h ) = k · O ( h log h ) and f ( h ) = O ( h ) , (2) in the third case, the compass ofthe wall W comes with a tree decomposition of width at most f ( k, h ) , and (3) the result is ac-companied by a 2 O h ( r ) · n time algorithm. Notice that a non-algorithmic version of this resultcould be indirectly derived, with worst functions, combining [31, Theorem 1.8] and the main resultof Kawarabayashi and Kobayashi in [26]. We present this result in this paper for the followingreasons: first because it is new, second because it is in a form suitable for future applications whereit is important that the compass has bounded treewidth, and third because its proof provides anindicative sample of the potential of the formalism of W -tilts that we suggest in ( β ) . This was first spotted in the conference article [39]. In fact, Theorem 5 applies not only to subwalls W of W , but also to every subwall W of the compass of W thatis not “contained” in a flap. See Subsection 2.3 for the details. δ ) An alternative concept of wall homogeneity. As mentioned before, the Flat Wall The-orem has been the combinatorial base for the
FPT -algorithms of [36] for
Minor Testing and
Disjoint Paths . One of the cornerstone ideas of [36] was to prove that the existence of a “bigenough” flat wall W in the input graph G implies that a minor-model of H or a collection of k disjoint paths in G can be safely rerouted so to avoid the central vertices of this wall (see Figure 1for a visualization of the central vertices of a wall). This permits us to declare parts of the wall“irrelevant” and find an equivalent instance of the problem with fewer vertices. In fact, avoidingthe central vertices is not so straightforward when dealing with a flat wall W. This is because thererouting has to be done inside the compass K of W where the paths should be rerouted throughdifferent, however “equivalent”, flaps of the compass. To deal with this, Robertson and Seymourdefined in [36] the concept of wall homogeneity . Roughly speaking, when a wall is homogenous thenthe variety of the ways that paths may be routed through the flaps that are inside some “brick” ofthe wall is the same for all bricks. In [36] it was proved that every big enough flat wall contains astill big homogeneous subwall where the claimed rerouting is possible, with the help of later resultsof the Graph Minors series [37, 38].The definition of wall homogeneity in [36] was based on the concept of the vision of a flap andwas quite particular to the problems it was dealing with. To our knowledge, after [36], no much useof homogeneity, as defined in [36], was done for algorithmic purposes. Most of the results where theirrelevant vertex technique was applied concerned questions on surface-embeddable graphs wherethe wall is “already” disk-embedded and there is no need of homogeneity (see e.g. [16, 21–24, 28–30,33, 34]). An indicative exception to this rule is the celebrated algorithm in [17, 18] for the problemof checking whether H is a topological minor of a graph G where some notion of homogeneity,tailor-made for this problem, was introduced (see [18, Theorem 5.8] and also [12]).In this paper we introduce an alternative notion of wall homogeneity that is simpler and moreversatile to use. This is done in Subsection 3.3 and is based on the framework introduced in ( β ) .Our definition may help dealing with the wide variety of the problems as it permits any versionof finite index flap equivalency (for instance, flap equivalency based on MSOL -expressibility). Weaccompany the definition with an
FPT -algorithm that finds a homogeneous subwall. This, togetherwith the main result of ( γ ) , can permit us to find “big-enough” homogeneous walls with compassesof bounded treewidth. This, in turn, will permit the answer of MSOL -queries in parts of the compassand will allow more elaborated applications of the irrelevant vertex technique (such as those usedfor problems on surface-embeddable graphs in [11, 15]). ( ε ) Regular flatness pairs and plane representations. We call a pair ( W, R ) flatness pair if W is a flat wall whose flatness is certified by the rendition R . Based on the framework of ( β ),in Subsection 3.4 we introduce a notion of regularity for flatness pairs, which roughly demandsthat the branching vertices of the wall are “internal” with respect to the flaps of the compassof W. Regular flatness pairs permit the representation of the compass of a flat wall by a graphembedded in a disk and a “well-arranged” wall inside it. This “plane” representation of flat wallswill appear handy in other applications. For instance, it has been a useful tool for the proofs ofthe main combinatorial results of [5, 39] as it makes it possible to translate routing questions insidecompasses to analogous questions on planar embeddings and deal with them in a more easy way(using the new homogeneity concept of ( δ ) ). 6 .3 Organization of the paper In Section 2 we provide some definitions and preliminary results and we state the two main resultsof this paper (Theorem 5 and Theorem 6), that assert the existence of an algorithm computinga tilt of a subwall of a flat wall and of an algorithm, that given a flatness pair outputs a regularflatness pair, respectively. We prove Theorem 5 and Theorem 6 in Section 4. In Section 3, wedevelop the tools to address the topics ( β ) , ( γ ) , ( δ ) , and ( ε ) listed above. Sets and integers.
We denote by N the set of non-negative integers. Given two integers p, q, where p ≤ q, we denote by [ p, q ] the set { p, . . . , q } . For an integer p ≥ , we set [ p ] = [1 , p ] and N ≥ p = N \ [0 , p − . For a set S, we denote by 2 S the set of all subsets of S and by (cid:0) S (cid:1) the set ofall subsets of S of size 2 . If S is a collection of objects where the operation ∪ is defined, then wedenote SSSSSSSSS S = S X ∈S X. Basic concepts on graphs.
As a graph G we denote any pair ( V, E ) where V is a finite setand E ⊆ (cid:0) V (cid:1) , that is, all graphs of this paper are undirected, finite, and without loops or multipleedges. We also define V ( G ) = V and E ( G ) = E. We say that a pair (
L, R ) ∈ V ( G ) × V ( G ) isa separation of G if L ∪ R = V ( G ) and there is no edge in G between L \ R and R \ L. Given avertex v ∈ V ( G ) , we denote by N G ( v ) the set of vertices of G that are adjacent to v in G. Also,given a set S ⊆ V ( G ) , we set N G ( S ) = S v ∈ S N G ( v ) . A vertex v ∈ V ( G ) is isolated if N G ( v ) = ∅ . For S ⊆ V ( G ) , we set G [ S ] = ( S, E ∩ (cid:0) S (cid:1) ) and use G \ S to denote G [ V ( G ) \ S ] . Given an edge e = { u, v } ∈ E ( G ) , we define the subdivision of e to be the operation of deleting e, adding a newvertex w, and making it adjacent to u and v. Given two graphs
H, G, we say that H is a subdivision of G if H can be obtained from G by subdividing edges. The contraction of an edge e = { u, v } ofa simple graph G results in a simple graph G obtained from G \ { u, v } by adding a new vertex uv adjacent to all the vertices in the set ( N G ( u ) ∪ N G ( v )) \ { u, v } . A graph G is a minor of a graph G if G can be obtained from a subgraph of G after a series of edge contractions. Disk-embedded graphs. A closed (resp. open ) disk is a set homeomorphic to the set { ( x, y ) ∈ R | x + y ≤ } (resp. { ( x, y ) ∈ R | x + y < } ). Let ∆ be a closed disk. We use bd (∆)to denote the boundary of ∆ and int (∆) to denote the open disk ∆ \ bd (∆) . When we embed agraph G in the plane or in a disk, we treat G as a set of points. This permits us to make setoperations operations between graphs and sets of points. We say that a graph G is ∆ -embedded if G is embedded in ∆ without crossings such that the intersection of bd (∆) and G (seen as a set ofpoints of ∆) is a subset of V ( G ) . A circle of ∆ is any set homeomorphic to { ( x, y ) ∈ R | x + y = 1 } . Given two distinct points x, y ∈ D, an ( x, y ) -arc of D is any subset of D that is homeomorphic to the closed interval [0 , . Walls.
Let k, r ∈ N . The ( k × r ) -grid is the graph whose vertex set is [ k ] × [ r ] and two vertices( i, j ) and ( i , j ) are adjacent if | i − i | + | j − j | = 1 . An elementary r -wall , for some odd integer7 ≥ , is the graph obtained from a (2 r × r )-grid with vertices ( x, y ) ∈ [2 r ] × [ r ] , after the removalof the “vertical” edges { ( x, y ) , ( x, y + 1) } for odd x + y, and then the removal of all vertices ofdegree one. Notice that, as r ≥ , an elementary r -wall is a planar graph that has a unique (upto topological isomorphism) embedding in the plane R such that all its finite faces are incidentto exactly six edges. The perimeter of an elementary r -wall is the cycle bounding its infinite face,while the cycles bounding its finite faces are called bricks . Also, the vertices in the perimeter of anelementary r -wall that have degree two are called pegs , while the vertices (1 , , (2 , r ) , (2 r − , , and (2 r, r ) are called corners (notice that the corners are also pegs).Figure 1: A 15-wall. The 3-branch vertices are depicted in cyan except from the corner and thecentral vertices that are depicted in red and orange respectively.An r -wall is any graph W obtained from an elementary r -wall ¯ W after subdividing edges (seeFigure 1). A graph W is a wall if it is an r -wall for some odd r ≥ r as the height of W. Given a graph G, a wall of G is a subgraph of G that is a wall. We insist that, for every r -wall, the number r is always odd.We call the vertices of degree three of a wall W . A cycle of W is a brick (resp.the perimeter ) of W if its 3-branch vertices are the vertices of a brick (resp. the perimeter) of ¯ W .
We denote by C ( W ) the set of all cycles of W. We use D ( W ) in order to denote the perimeter ofthe wall W. A brick of W is internal if it is disjoint from D ( W ) . Subwalls.
Given an elementary r -wall ¯ W , some i ∈ { , , . . . , r − } , and i = ( i + 1) / , the i -thvertical path of ¯ W is the one whose vertices, in order of appearance, are ( i, , ( i, , ( i + 1 , , ( i +1 , , ( i, , ( i, , ( i + 1 , , ( i + 1 , , ( i, , . . . , ( i, r − , ( i, r − , ( i + 1 , r − , ( i + 1 , r ) . Also, givensome j ∈ [2 , r −
1] the j -th horizontal path of ¯ W is the one whose vertices, in order of appearance,are (1 , j ) , (2 , j ) , . . . , (2 r, j ) . A vertical (resp. horizontal ) path of W is one that is a subdivision of a vertical (resp. horizontal)path of ¯ W .
Notice that the perimeter of an r -wall W is uniquely defined regardless of the choice ofthe elementary r -wall ¯ W . A subwall of W is any subgraph W of W that is an r -wall, with r ≤ r, W are subpaths of the vertical (resp. horizontal)paths of W. Tilts.
The interior of a wall W is the graph obtained from W if we remove from it all edges of D ( W ) and all vertices of D ( W ) that have degree two in W. Given two walls W and ˜ W of a graph G, we say that ˜ W is a tilt of W if ˜ W and W have identical interiors. Paintings.
Let ∆ be a closed disk. Given a subset X of ∆ , we denote its closure by ¯ X and itsboundary by bd ( X ) . A ∆ -painting is a pair Γ = (
U, N ) where• N is a finite set of points of ∆ , • N ⊆ U ⊆ ∆ , and• U \ N has finitely many arcwise-connected components, called cells , where, for every cell c, ◦ the closure ¯ c of c is a closed disk and ◦ | ˜ c | ≤ , where ˜ c := bd ( c ) ∩ N. We use the notation U (Γ) := U, N (Γ) := N and denote the set of cells of Γ by C (Γ) . For convenience,we may assume that each cell of Γ is an open disk of ∆ . Notice that, given a ∆-painting Γ , the pair ( N (Γ) , { ˜ c | c ∈ C (Γ) } ) is a hypergraph whose hy-peredges have cardinality at most three and Γ can be seen as a plane embedding of this hypergraphin ∆ . Renditions.
Let G be a graph, and let Ω be a cyclic permutation of a subset of V ( G ) that wedenote by V (Ω) . By an Ω -rendition of G we mean a triple (Γ , σ, π ) , where(a) Γ is a ∆-painting for some closed disk ∆ , (b) π : N (Γ) → V ( G ) is an injection, and(c) σ assigns to each cell c ∈ C (Γ) a subgraph σ ( c ) of G, such that(1) G = S c ∈ C (Γ) σ ( c ) , (2) for distinct c, c ∈ C (Γ) , σ ( c ) and σ ( c ) are edge-disjoint,(3) for every cell c ∈ C (Γ) , π (˜ c ) ⊆ V ( σ ( c )) , (4) for every cell c ∈ C (Γ) , V ( σ ( c )) ∩ S c ∈ C (Γ) \{ c } V ( σ ( c )) ⊆ π (˜ c ) , and(5) π ( N (Γ) ∩ bd (∆)) = V (Ω) , such that the points in N (Γ) ∩ bd (∆) appear in bd (∆) in thesame ordering as their images, via π, in Ω . Given an Ω-rendition (Γ , σ, π ) of a graph G, we call a cell c of Γ trivial if π (˜ c ) = V ( σ ( c )) . We say that an Ω-rendition (Γ , σ, π ) of a graph G is tight if the following conditions are satisfied:9i) If there are two points x, y of N (Γ) such that e = { π ( x ) , π ( y ) } ∈ E ( G ) , then there is a cell c ∈ C (Γ) such that σ ( c ) is the two-vertex connected graph ( e, { e } ) , (ii) for every c ∈ C (Γ) , every two vertices in π (˜ c ) belong to some path of σ ( c ) , (iii) for every c ∈ C (Γ) and every connected component C of the graph σ ( c ) \ π (˜ c ) , if N σ ( c ) ( V ( C )) = ∅ , then N σ ( c ) ( V ( C )) = π (˜ c ) , (iv) there are no two distinct non-trivial cells c and c such that π ( ˜ c ) = π ( ˜ c ) , and(v) for every c ∈ C (Γ) there are | ˜ c | vertex-disjoint paths in G from π (˜ c ) to the set V (Ω) . (i) (ii)(iii) (iv)(v) Figure 2: A graph G together with an Ω-rendition of G, where all tightness conditions are violated. Lemma 3.
There is a linear-time algorithm that, given a graph G and an Ω -rendition (Γ , σ, π ) of G, outputs a tight Ω -rendition of G. Proof.
We argue about how to transform (Γ , σ, π ) to a tight Ω-rendition of G in O ( n + m ) time.See Figure 2 for an example of a graph G together with an Ω-rendition of G that violates each ofthe five tightness conditions (indicated in the figure).For the first property, let e = { π ( x ) , π ( y ) } ∈ E ( G ) be an edge of G that belongs to some σ ( c )with | V ( σ ( c )) | > . Then, we add a new cell c new to the rendition, where π (˜ c new ) = { π ( x ) , π ( y ) } and σ ( c new ) = ( e, { e } ) Also, we remove the edge e from σ ( c ) . For the second property, let c be a cell in C (Γ) and let C be a collection containing everycomponent of the graph σ ( c ) . We say that C , C ∈ C are equivalent if V ( C ) ∩ π (˜ c ) = V ( C ) ∩ π (˜ c ) . Notice that each equivalence class of this equivalence relation corresponds to some partition P of π (˜ c ) . If this equivalence relation has only one class, then (ii) holds, because of condition (c.3) of10he definition of rendition. If not, we remove c from the rendition and we replace it with as manycells as the number of equivalence classes, one for each equivalence class and we update σ so thateach new cell is mapped to the union of the members of the equivalence class corresponding to it.For the third property, consider some c ∈ C (Γ) , and observe that, because of (i) and (ii), thegraph σ ( c ) \ π (˜ c ) contains at least one connected component, say C ∗ , with N σ ( c ) ( V ( C )) = ∅ . Let C be a collection containing every component of the graph σ ( c ) \ π (˜ c ) . We say that C , C ∈ C are equivalent if N σ ( c ) ( V ( C )) = N σ ( c ) ( V ( C )) . Notice that this equivalence relation has at most eightequivalence classes, each corresponding to a subset of π (˜ c ) . For each subset X of π (˜ c ) , we definethe graph F X as the union of the graphs in the corresponding equivalence class. Let also X ∗ bethe non-empty subset of π (˜ c ) such that C ∗ is a subgraph of F X ∗ . We enhance F X ∗ := F X ∗ ∪ F ∅ . Wenow remove the cell c from the rendition and for every non-empty X ∈ π (˜ c ) where F X is non-null,we add a new cell c X and we update σ by mapping each c X to the graph F X . For property (iv), for every two distinct non-trivial cells c and c with π ( ˜ c ) = π ( ˜ c ) , we remove c from the rendition and we update σ ( c ) := σ ( c ) ∪ σ ( c ) . The last property can be achieved as follows: we first construct an auxiliary planar graph G by substituting in G each σ ( c ) by a clique on π (˜ c ) (that is a vertex, an edge, or a triangle) andby adding a new vertex v new adjacent to all the vertices in V (Ω); then the new rendition can beeasily constructed starting from the triconnected component C of G that contains v new (to find thetriconnected components, one may use the classic algorithm of Hopcroft and Tarjan [19] that runsin O ( n + m ) time) and then attaching to C, as images of the updated π, the other triconnectedcomponents.In the rest of this paper we use only conditions (i)–(iii) of the tightness definition. However, weadopt the above, more strict, version of tightness as it will be useful in further applications. Let W be an r -wall, for some odd integer r ≥ . We say that a pair (
P, C ) ⊆ D ( W ) × D ( W ) is a choice of pegs and corners for W if W is the subdivision of an elementary r -wall ¯ W where P and C are the pegs and the corners of ¯ W , respectively (clearly, C ⊆ P ). To get more intuition, noticethat a wall W can occur in several ways from the elementary wall ¯ W , depending on the way thevertices in the perimeter of ¯ W are subdivided. Each of them gives a different selection ( P, C ) ofpegs and corners of W. Let an odd integer r ≥ W be an r -wall of some graph G. We say that W is a flat r -wall of G if there is a separation ( X, Y ) of G and a choice ( P, C ) of pegs and corners for W such that:• V ( W ) ⊆ Y, • P ⊆ X ∩ Y ⊆ V ( D ( W )) , and• if Ω is the cyclic ordering of the vertices X ∩ Y as they appear in D ( W ) , then there exists anΩ-rendition (Γ , σ, π ) of G [ Y ] . Because of Lemma 3, we can assume (and we also demand) that the Ω-rendition (Γ , σ, π ) of G [ Y ] in the above definition is always tight. We mention here that Chuzhoy [7] uses a slightly11ifferent notion of flatness, where the separation ( X, Y ) consists of two edge-disjoint subgraphs ,instead of two vertex sets , and where the graph Y may play the role of the compass. Flatness pairs.
Given the above, we say that the choice of the 7-tuple R = ( X, Y, P, C, Γ , σ, π ) certifies that W is a flat wall of G . We call the pair ( W, R ) a flatness pair of G and define the height of the pair ( W, R ) to be the height of W. We use the term cell of R in order to refer to thecells of Γ . We call the graph G [ Y ] the R -compass of W in G, denoted by compass R ( W ) . We define the flaps of the wall W in R as flaps R ( W ) := { σ ( c ) | c ∈ C (Γ) } . Given a flap F ∈ flaps R ( W ) , we defineits base as ∂F := V ( F ) ∩ π ( N (Γ)) . A flap F ∈ flaps R ( W ) is trivial if | ∂F | = 2 and F consistsof one edge between the two vertices in ∂F. We call the edges of the trivial flaps short edges of compass R ( W ). A cell c of R is untidy if π (˜ c ) contains a vertex x of W such that two of the edgesof W that are incident to x are edges of σ ( c ) . Notice that if c is untidy then | ˜ c | = 3 . Figure 3: A flat 7-wall W in a graph G whose flatness is certified by some rendition R wherethe choice of pegs and corners in R corresponds to the squared vertices. We depict only the R -compass of W that consists of W and some “black paths” between the vertices of W. The 5-wall˜ W consisting of the fat edges (purple, green, blue) is a flat R -normal wall of compass R ( W ) . Theflatness of ˜ W is certified by the rendition ˜ R = ( X , Y , P , C , Γ , σ , π ) , where X contains all thevertices incident to at least one orange edge plus the non-depicted vertices in the grey area, Y contains all vertices that are either in a “fat” black path or incident to at least two fat edges, thepegs are the diamond vertices, and the corners are the fat diamond vertices (that are also pegs).For the (tight) Ω -rendition (Γ , σ , π ) of G [ Y ] , see Figure 4.In Figure 3 we depict a flat wall W in a graph G as well as the R -compass of W in G, for somerendition R certifying its flatness. Notice that there is a unique subwall W of W that is disjoint12rom D ( W ) and has height five. Interestingly, the subwall W is not a flat wall of G, however thereis a tilt ˜ W of W that is a flat wall of G. The wall ˜ W is depicted in Figure 3 and the renditioncertifying its flatness is depicted in Figure 4.Figure 4: The painting of the rendition ˜ R certifying the flatness of the 5-wall ˜ W of Figure 3. The˜ R -compass of ˜ W has two types of flaps: those whose base has three vertices (they are images ofthe blue cells) and those that are trivial (they are images of the purple cells). Cell classification.
Given a cycle C of compass R ( W ) , we say that C is R -normal if it is nota subgraph of a flap F ∈ flaps R ( W ) . Given an R -normal cycle C of compass R ( W ) , we call a cell c of R C -perimetric if σ ( c ) contains some edge of C. Notice that if c is C -perimetric, then π (˜ c )contains two points p, q ∈ N (Γ) such that π ( p ) and π ( q ) are vertices of C where one, say P in c , of thetwo ( π ( p ) , π ( q ))-subpaths of C is a subgraph of σ ( c ) and the other, denoted by P out c , ( π ( p ) , π ( q ))-subpath contains at most one internal vertex of σ ( c ) , which should be the (unique) vertex z in ∂σ ( c ) \ { π ( p ) , π ( q ) } . We pick a ( p, q )-arc A c in ˆ c := c ∪ ˜ c such that π − ( z ) ∈ A c if and only if P in c contains the vertex z as an internal vertex.We consider the circle K C = SSSSSSSSS { A c | c is a C -perimetric cell of R } and we denote by ∆ C theclosed disk bounded by K C that is contained in ∆ . A cell c of R is called C -internal if c ⊆ ∆ C and is called C -external if ∆ C ∩ c = ∅ . Notice that the cells of R are partitioned into C -internal, C -perimetric, and C -external cells.Let c be a tidy C -perimetric cell of R where | ˜ c | = 3 . Notice that c \ A c has two arcwise-connectedcomponents and one of them is an open disk D c that is a subset of ∆ C . If the closure D c of D c contains only two points of ˜ c then we call the cell c C -marginal . Influence.
For every R -normal cycle C of compass R ( W ) we define the set influence R ( C ) = { σ ( c ) | c is a cell of R that is not C -external } . W in a graph G, the painting of a rendition R certifying its flatness, a subwall W of W, of height three, which is R -normal, and the R -flaps of W, that correspond to either W -perimetric (depicted in grey) or W -internal cells (depicted in green). The circle K W is the fatorange cycle. The W -marginal cells are depicted in light grey and the untidy cells are those withdashed boundary.A wall W of compass R ( W ) is R -normal if D ( W ) is R -normal . Notice that every wall of W (and hence every subwall of W ) is an R -normal wall of compass R ( W ) . We denote by S R ( W ) theset of all R -normal walls of compass R ( W ) . Given a W ∈ S R ( W ) and a cell c of R we say that c is W -perimetric/internal/external/marginal if c is D ( W )-perimetric/internal/external/marginal.We also use K W , ∆ W , influence R ( W ) as shortcuts for K D ( W ) , ∆ D ( W ) , influence R ( D ( W )) . Regular pairs.
Let ( W, R ) be a flatness pair of a graph G. We call a flatness pair ( W, R ) of agraph G regular if none of its cells is W -external, W -marginal, or untidy. Tilts of flatness pairs.
Let ( W, R ) and ( ˜ W , ˜ R ) be two flatness pairs of a graph G and let W ∈ S R ( W ) . We also assume that R = ( X, Y, P, C, Γ , σ, π ) and ˜ R = ( X , Y , P , C , Γ , σ , π ) . Wesay that ( ˜ W , ˜ R ) is a W -tilt of ( W, R ) if• ˜ R does not have ˜ W -external cells,• ˜ W is a tilt of W ,
14 the set of ˜ W -internal cells of ˜ R is the same as the set of W -internal cells of R and theirimages via σ and σ are also the same,• compass ˜ R ( ˜ W ) is a subgraph of SSSSSSSSS influence R ( W ) , and• if c is a cell in C (Γ ) \ C (Γ) , then | ˜ c | ≤ . The next observation follows from the definitions of regular flatness pairs and tilts.
Observation . If ( W, R ) is a regular flatness pair, then for every W ∈ S R ( W ) every W -tilt of( W, R ) is also regular.The main results of this paper are the following. Theorem 5.
There exists an algorithm that given a graph G, a flatness pair ( W, R ) of G, and awall W ∈ S R ( W ) , outputs a W -tilt of ( W, R ) in O ( n + m ) time. Theorem 6.
There is an algorithm that, given a graph G and a flatness pair ( W, R ) of G, outputsa regular flatness pair ( W ? , R ? ) of G, with the same height as ( W, R ) such that compass R ? ( W ? ) ⊆ compass R ( W ) . This algorithm runs in O ( n + m ) time. In this section we apply Theorem 5 and Theorem 6 in order to address the items ( β ) , ( γ ) , ( δ ) ,and ( ε ) discussed in the introduction. We present the following result from [31], stated in our new framework.
Proposition 7.
There are two functions f : N → N and f : N → N and an algorithm that receivesas input a graph G, an odd integer r ≥ , a t ∈ N ≥ , and an f ( t ) · r -wall W in G, and outputs, in O t ( n ) time, • either that K t is a minor of G or • a set A ⊆ V ( G ) where | A | ≤ f ( t ) and a flatness pair ( ˜ W , ˜ R ) of G \ A of height r, such that ˜ W is a tilt of a subwall W of W. Moreover f ( t ) = O ( t ) and f ( t ) = O ( t ) . An alternative of the above where f ( t ) = O ( t ) and f ( t ) = t − O ( t ) has been proved byChuzhoy in [7] with a running time that is polynomial in the input size. However, we prefer theversion of Kawarabayashi, Thomas, and Wollan [31] as their algorithm is linear.15 .2 Apex-walls with compasses of bounded treewidth We first define the notion of treewidth. A tree decomposition of a graph G is a pair ( T, χ ) where T is a tree and χ : V ( T ) → V ( G ) such that1. S t ∈ V ( T ) χ ( t ) = V ( G ) ,
2. for every edge e of G there is a t ∈ V ( T ) such that χ ( t ) contains both endpoints of e, and3. for every v ∈ V ( G ) , the subgraph of T induced by { t ∈ V ( T ) | v ∈ χ ( t ) } is connected.The width of ( T, χ ) is defined as w ( T, χ ) := max (cid:8) | χ ( t ) | − (cid:12)(cid:12) t ∈ V ( T ) (cid:9) . The treewidth of G isdefined as tw ( G ) := min (cid:8) w ( T, χ ) (cid:12)(cid:12) ( T, χ ) is a tree decomposition of G (cid:9) . This subsection is dedicated to the proof of the following result.
Theorem 8.
There is a function f : N → N and an algorithm that receives as input a graph G, an odd integer r ≥ , and a t ∈ N ≥ , and outputs, in O t ( r ) · n time, one of the following: • a report that K t is a minor of G, • a tree decomposition of G of width at most f ( t ) · r, or • a set A ⊆ V ( G ) , where | A | ≤ f ( t ) , a regular flatness pair ( W, R ) of G \ A of height r, and atree decomposition of the R -compass of W of width at most f ( t ) · r. (Here f ( t ) is the functionof Proposition 7 and f ( t ) = 2 O ( t log t ) . ) We will need some additional results in order to prove Theorem 8. First we need the followingresult that is derived from [35]. For a detailed analysis of the results of [35], see [3].
Proposition 9.
There exists an algorithm with the following specifications:
Input : A graph G and a non-negative integer k such that | V ( G ) | ≥ k . Output : A graph G ∗ such that | V ( G ∗ ) | ≤ (1 − k ) · | V ( G ) | and: • Either G ∗ is a subgraph of G such that tw ( G ) = tw ( G ∗ ) , or • G ∗ is obtained from G after identifying the vertices of a matching in G. Moreover, this algorithm runs in O ( k ) · n time. The following result of Kawarabayashi and Kobayashi [26], provides a linear relation betweenthe treewidth and the height of a largest wall in a minor-free graph.
Proposition 10.
There is a function f : N → N such that, for every t, r ∈ N and every graph G that does not contain K t as a minor, if tw ( G ) ≥ f ( t ) · r, then G contains an r -wall. In particular,one may choose f ( t ) = 2 O ( t log t ) . The following is the main result of [6]. We will use it to compute a tree decomposition of agraph of bounded treewidth. 16 roposition 11.
There is an algorithm that, given a graph G and an integer k, outputs eithera report that tw ( G ) > k, or a tree decomposition of G of width at most k + 4 . Moreover, thisalgorithm runs in O ( k ) · n time. The following result is derived from [1]. We will use it in order to find a wall in a graph ofbounded treewidth, given a tree decomposition of it.
Proposition 12.
There is an algorithm that, given a graph G, a graph H on h edges withoutisolated vertices, and a tree decomposition of G of width at most k, outputs, if it exists, a minor of G isomorphic to H. Moreover, this algorithm runs in O ( k log k ) · h O ( k ) · O ( h ) · m time. We start by proving the following “light version” of Theorem 8.
Lemma 13.
There exists an algorithm as follows:
Find-Wall ( G, t, r ) Input : A graph G, an odd r ∈ N ≥ , and a t ∈ N ≥ . Output : One of the following: • a report that K t is a minor of G, • a report that G has treewidth at most f ( t ) · r, where f is as in Proposition 10, or • an r -wall W of G. Moreover, this algorithm runs in O t ( r ) · n time.Proof. We set c := f ( t ) · r. We now describe a recursive algorithm as follows.We first argue for the base case, namely when | V ( G ) | < c . To check whether K t is aminor of G, we use the minor-containment algorithm of Robertson and Seymour [36], which runsin O t ( | V ( G ) | ) = O t ( r ) time, and if this is the case, we report the same and stop. If not, then wecheck whether tw ( G ) ≤ c, using the algorithm of Arnborg, Corneil, and Proskurowski [4], in time O ( | V ( G ) | c +2 ) = 2 O t ( r log r ) , and if this is the case, we report the same and stop. If not, we deal withthe case where G does not contain K t as a minor and tw ( G ) > c. By Proposition 10 we know that G contains an r -wall. To find such a wall, we first consider an arbitrary ordering ( v , . . . , v | V ( G ) | )of the vertices of G. For each i ∈ [ | V ( G ) | ] , we set G i to be the graph induced by the vertices v , . . . , v i . We iteratively run the algorithm of Proposition 11 on G i and c for ascending values of i. This algorithm runs in 2 O ( c ) · | V ( G ) | = 2 O t ( r +log r ) time. Let j ∈ [ | V ( G ) | ] be the smallest integersuch that the above algorithm outputs a report that tw ( G j ) > c and notice that there exists a treedecomposition ( T j , χ j ) of G j (obtained by the one of G j − by adding the vertex v j in the appropriatebags) of width at most 5 c + 5 . The fact that G j does not contain K t as a minor and tw ( G j ) > c, implies that G j contains an r -wall W, that is also a wall of G. To detect W, we run the algorithm ofProposition 12 on G j , W, and ( T j , χ j ) . This algorithm runs in 2 O t ( r ) · | V ( G ) | = 2 O t ( r +log r ) time.Therefore, in the case where | V ( G ) | ≤ c , we obtain one of the three possible outputs in time2 O t ( r ) . If | V ( G ) | ≥ c , then we call the algorithm of Proposition 9 with input ( G, c ) , which outputsa graph G ∗ such that | V ( G ∗ ) | ≤ (1 − c ) · | V ( G ) | and17 . either G ∗ is a subgraph of G such that tw ( G ) = tw ( G ∗ ) , or B . G ∗ is obtained from G after identifying the vertices of a matching M of G. In both cases, we recursively call the algorithm on G ∗ and we distinguish the following two cases. Case A : G ∗ is a subgraph of G such that tw ( G ) = tw ( G ∗ ) . If the recursive call on G ∗ reports that K t is a minor of G ∗ , then we report the same for G as well. If the recursive call on G ∗ reports that tw ( G ∗ ) ≤ c, then we return that tw ( G ) ≤ c. If it outputs an r -wall W of G ∗ , then we return W asa wall of G. Case B : G ∗ is obtained from G after contacting the edges of a matching of G. If the recursive call on G ∗ reports that tw ( G ∗ ) ≤ c, then we do the following. We first notice thatthe fact that tw ( G ∗ ) ≤ c implies that tw ( G ) ≤ c, since we can obtain a tree decomposition ( T , χ )of G from a tree decomposition ( T ∗ , χ ∗ ) of G ∗ , by replacing, in every t ∈ T ∗ , every occurrence ofa vertex of G ∗ that is a result of an edge contraction by its endpoints in G. Thus, we can call thealgorithm of Proposition 12 on
G, K t , and ( T , χ ) in order to check whether G contains K t as aminor in 2 O t ( r log r ) · n steps and if this is the case, we report the same and stop (keep in mind that c = O t ( r )). If not, then using the same algorithm we can also find in G, if it exists, an r -wall W asa minor in 2 O t ( r ) · n time and, if this is the case, we report the same and stop. In the remainingcase, we can safely report, because of Proposition 10, that tw ( G ) ≤ f ( t ) · r = c. If the recursive call on G ∗ outputs an r -wall W ∗ of G ∗ , then by uncontracting the edges of M in W ∗ we can also return an r -wall of G. Finally, if the output is that K t is a minor of G ∗ , then wereturn that the same holds for G. It is easy to see that the running time of the above algorithm is T ( n, r, t ) ≤ T ((1 − c ) · n, r, t ) + 2 O t ( r ) · n, which implies that T ( n, r, t ) = 2 O t ( r ) · n, as claimed.Given a flatness pair ( W, R ) of a graph G and a set L ⊆ V ( G ) , we say that ( W, R ) is L -avoiding if L ∩ V ( compass R ( W )) = ∅ . We now proceed to the proof of Theorem 8.
Proof of Theorem 8.
Notice that there is a constant c t , depending on t, such that if | E ( G ) | >c t ·| V ( G ) | , then G contains K t as a minor [40]. We therefore assume that | E ( G ) | = O t ( n ) , otherwisewe can immediately report that K t is a minor of G and stop. We first give an algorithm with thefollowing specifications. This algorithm involves recursion assuming an input with an additionalset L that should be avoided by the desired flatness pair. For notational convenience, we define z : N → N as z ( r, t ) = 2 · ( d p f ( t ) + 2 e + 1) · f ( t ) · ( f ( t ) + 1) · ( r + 2) . Algorithm
Find_Low_TW_compass ( G, r, t, L ) . Input : an odd r ∈ N ≥ , a t ∈ N ≥ , a graph G where tw ( G ) > z ( r, t ) , and a set L ⊆ V ( G ) where | L | ≤ f ( t ) + 1 . Output : either a report that K t is a minor of G or a set A ⊆ V ( G ), where | A | ≤ f ( t ) , an L -avoiding flatness pair ( W, R ) of G \ A of height f ( t ), and a tree decomposition of the R -compassof W of width at most 5 · z ( r, t ) + 4 . tep 1 . We set ‘ as the smallest odd integer that is not smaller than p f ( t ) + 2 . Also, let ˜ f ( t ) bethe smallest odd integer that is not smaller than f ( t ) . These augmentations are necessary in orderto guarantee that the considered subwalls will be of odd height. We also set r = 2 · ( r + 2) + 1 . Runthe algorithm of Lemma 13 for
G, ‘ · ˜ f ( t ) · r , and t. This takes 2 O t ( r ) · n time. If the output is a reportthat K t is a minor of G, then return the same. Otherwise, because, tw ( G ) > z ( r, t ) ≥ ‘ · f ( t ) · ˜ f ( t ) · r , the algorithm returns an ‘ · ˜ f ( t ) · (2( r + 2) + 1)-wall W of G. Step 2 . Call the algorithm of Proposition 7 on
G, ‘ · r , t, and W. This takes O t ( n ) time, since | E ( G ) | = O t ( n ) . If the output is a report that K t is a minor of G, then return the same. Otherwise,we have a set A ⊆ V ( G ), where | A | ≤ f ( t ) , and a flatness pair ( ˜ W , ˜ R ) of G \ A of height ‘ · r . Step 3 . Let W be a subwall of ˜ W of height r such that none of the vertices in L belongs to influence ˜ R ( W ) . The subwall W exists because ‘ ≥ f ( t ) + 2 ≥ | L | + 1 and ˜ W has height ‘ · r . We also consider four pairwise disjoint ( r + 2)-subwalls of W , namely W , W , W , and W , andobserve that each W i is also a subwall of ˜ W . For every i ∈ [4] , we call the algorithm of Theorem 5on G \ A, ( ˜ W , ˜ R ) , and W i which outputs, in O t ( n ) time, a W i -tilt ( ˜ W i , ˜ R i ) of ( ˜ W , ˜ R ) . Let K i bethe compass of ˜ W i in ˜ R i . We finally fix i so that ˜ W i is a wall among W , W , W , and W where | V ( K i ) | is minimized. Observe that | V ( K i ) | ≤ | V ( G ) | / W i , ˜ R i ) is L -avoiding. Indeed,since ( ˜ W i , ˜ R i ) is a W i -tilt of ( ˜ W , ˜ R ) , K i = compass ˜ R i ( ˜ W i ) is a subgraph of SSSSSSSSS influence ˜ R ( W i ) that,in turn, is a subgraph of SSSSSSSSS influence ˜ R ( W ) and by definition of W , influence ˜ R ( W ) ∩ L = ∅ . We update W ← ˜ W i , R ← ˜ R i and we set K = compass R ( W ) . Recall that ( W, R ) is an L -avoiding flatness pair of G \ A of height r + 2 . Step 4 . We now consider the subwall W of W obtained from W \ D ( W ) after repeatedly removingvertices of degree one until no such vertices exist anymore. Notice that W is an r -wall of G \ A. Wecall the algorithm of Theorem 5 on G \ A, ( W, R ) , and W which outputs, in O t ( n ) time, a W -tilt( ˜ W , ˜ R ) of ( W, R ) . Let K be the ˜ R -compass of ˜ W . Clearly, ( ˜ W , ˜ R ) is L -avoiding as well. Step 5 . Let G D be the graph obtained from G [ V ( K ) ∪ A ] if we contract all the vertices of D ( W )to a single vertex v ∗ . Since ( ˜ W , ˜ R ) is a W -tilt of ( W, R ) , K = compass ˜ R ( ˜ W ) is a subgraph of SSSSSSSSS influence R ( W ) , and therefore the perimeter of W and the graph K do not have any vertex incommon. This implies that K is a subgraph of G D . Step 6 . Call the algorithm of Proposition 11 with input G D and z ( r, t ) . This runs in 2 O t ( r ) · n time. If the output is a tree decomposition of G D of width at most 5 · z ( r, t ) + 4 , then, as K is asubgraph of G D , we have that ( ˜ W , ˜ R ) is an L -avoiding flatness pair of G \ A of height r where the˜ R -compass of ˜ W has treewidth at most 5 · z ( r, t ) + 4 . In this case, the algorithm outputs the pair( ˜ W , ˜ R ) and the corresponding tree decomposition of the ˜ R -compass K of ˜ W obtained from theone of G D by removing the vertices in V ( G D ) \ V ( K ) . Step 7 . Suppose now that tw ( G D ) > z ( r, t ) . Notice that, by construction, if G D \ A hasan { v ∗ } -avoiding flatness pair ( W ∗ , R ∗ ) of height r, then ( W ∗ , R ∗ ) will also be an L -avoidingflatness pair of G \ A. Moreover, since G D is a minor of G, if G D contains K t as a minorthen also G does. Notice also that | A ∪ { v }| ≤ f ( t ) + 1 . Therefore, we can safely return
Find_Low_TW_compass ( G D , r, t, A ∪ { v } ) . This completes the description of the algorithmand its correctness.Notice that the running time of the above algorithm is T ( n, r, t ) ≤ T ( n/ f ( t ) , r, t )+2 O t ( r ) · n, which implies that T ( n, r, t ) = 2 O t ( r ) · n.
19e define the function f : N → N so that f ( t ) = min { c ∈ N | ∀ r ≥ , · z ( r, t ) + 4 ≤ c · r } . The algorithm claimed by the theorem calls first the algorithm of Proposition 11 with input G and z ( r, t ) . This runs in 2 O t ( r ) · n time. If the output is a tree decomposition of G of width atmost 5 · z ( r, t ) + 4 ≤ f ( t ) · r, then we report this and we are done. If the output is a report that tw ( G ) > z ( r, t ) , then we run Algorithm Find_Low_TW_compass ( G, r, t, L ) for L = ∅ . Thismay provide either a report that K t is a minor of G, or a set A ⊆ V ( G ), where | A | ≤ f ( t ) , aflatness pair ( W, R ) of G \ A of height f ( t ) that can be made regular by Theorem 6, and a treedecomposition of the R -compass of W of width at most 5 · z ( r, t ) + 4 ≤ f ( t ) · r, and these are thepossible outputs of the claimed algorithm. Palettes and homogeneity.
Let w ∈ N , let G be a graph, and let ( W, R ) be a flatness pairof G. A flap-coloring of ( W, R ) with w colors is any function ζ : flaps R ( W ) → [ w ] . For every R -normal cycle C of compass R ( W ) , we define ζ -palette ( C ) = { ζ ( F ) | F ∈ influence R ( C ) } . We saythat the flatness pair ( W, R ) of G is ζ -homogeneous if every internal brick of W (seen as a cycle of compass R ( W )) has the same ζ -palette .Finding a homogeneous flatness pair inside a flatness pair has a price which is determined bythe following lemma. Lemma 14.
There is a function f : N → N , whose images are odd integers, such that for every w ∈ N ≥ and every odd integer r ≥ , if G is a graph, ( W, R ) is a flatness pair of G of height f ( r, w ) , and ζ is a flap-coloring of ( W, R ) with w colors, then W contains some subwall W ofheight r such that every W -tilt of ( W, R ) is ζ -homogeneous. Moreover, f ( r, w ) = O ( r w ) . Proof.
Let w ∈ N and an odd integer r ≥ . We define the function f : N → N so that, for every x ∈ N , f ( x,
1) = x while, for y ≥ , we set f ( x, y ) = x · ( f ( x, y − −
1) + 1 . Notice that if x isodd, then f ( x, y ) is also odd for every y ∈ N ≥ . Let G be a graph, ( W, R ) be a flatness pair of G of height f ( r, w ) , and ζ be a flap-coloring of( W, R ) with w colors. We prove the lemma by induction on w. Clearly, if w = 1 , then the lemmaholds trivially as, in this case, for every brick B of W, ζ -palette ( B ) = { } , and therefore as W is a subwall of itself, every W -tilt of ( W, R ) is a flatness pair of G of height f ( r,
1) = r that is ζ -homogeneous.Suppose now that w ≥ w. We set q = f ( r, w − . We define the subwall W of W by taking the union of the i -th horizontal and the i -th vertical paths of W for all i ∈ { j · ( q −
1) + 1 | j ∈ [ r ] } . If for every brick B of W it holds that ζ -palette ( B ) = [ w ] , then consider a W -tilt ( ˜ W , ˜ R ) of ( W, R ) . The third property in the definitionof a tilt of a flatness pair implies that for every internal brick ˜ B of ˜ W there is an internal brick B of W such that influence R ( B ) = influence ˜ R ( ˜ B ) . Therefore, for every internal brick ˜ B of ˜ W ,ζ -palette ( ˜ B ) = [ w ] . Therefore, ( ˜ W , ˜ R ) is a flatness pair of G of height r that is ζ -homogeneous.Otherwise, let ˘ B be some brick of W such that | ζ -palette ( ˘ B ) | < w. Notice that ˘ B is the perimeterof a subwall ˘ W of W of height q. From the induction hypothesis applied to ˘
W , we have that ˘ W hasa subwall W (that is a subwall of W as well) such that every W -tilt of ( W, R ) is a flatness pair of G of height r that is ζ -homogeneous. The lemma follows by observing that f ( r, w ) = O ( r w ) .
20e now prove the main result of this subsection.
Lemma 15.
There is an algorithm that receives as input w ∈ N ≥ , an odd integer r ≥ , a graph G, a flatness pair ( W, R ) of G of height f ( r, w ) , and a flap-coloring ζ of ( W, R ) with w colors, andoutputs a ζ -homogeneous flatness pair ( ˘ W , ˘ R ) of G of height r that is a W -tilt of ( W, R ) for somesubwall W of W. This algorithm runs in time O ( wr log r ) · ( n + m ) .Proof. Let W be the collection of all r -subwalls of W. Clearly |W| = (cid:0) f ( r,w ) r (cid:1) = 2 O ( wr log r ) . Foreach W ∈ W , we call the algorithm of Theorem 5 on G, ( W, R ) , and W , which outputs, a W -tilt( ˜ W , ˜ R ) of ( W, R ) . This algorithm runs in O ( n + m ) time. Then, for every W ∈ W , we checkwhether ( ˜ W , ˜ R ) is ζ -homogeneous by computing the ζ - palette ( ˜ B ) for every internal brick ˜ B of ˜ W . This is done in linear time. Lemma 14 guarantees that since the height of ( W, R ) is f ( r, w ) , W contains a subwall W of height r such that every W -tilt of ( W, R ) is ζ -homogeneous. Therefore,the above procedure will detect a flatness pair ( ˜ W , ˜ R ) of G that is ζ -homogeneous and has height r, which we return. Let G be a graph and let ( W, R ) be a flatness pair of G. Let also R = ( X, Y, P, C, Γ , σ, π ) , where(Γ , σ, π ) is an Ω-rendition of G [ Y ] and Γ = ( U, N ) is a ∆-painting. The ground set of W in R is ground R ( W ) := π ( N (Γ)) and we refer to the vertices of this set as the ground vertices of the R -compass of W in G. Notice that ground R ( W ) may contain vertices of compass R ( W ) that are notnecessarily vertices of W. For instance, in Figure 3, all the ground vertices of the ˜ R -compass of ˜ W are vertices of ˜ W , while in Figure 5, there are ground vertices of the R -compass of W that are notvertices of W. Figure 6: The ˜ R -leveling of the flat 5-wall ˜ W of Figure 3.We define the R - leveling of W in G, denoted by W R , as the bipartite graph where one part isthe ground set of W in R , the other part is a set vflaps R ( W ) = { v F | F ∈ flaps R ( W ) } containing21ne new vertex v F for each flap F of W in R , and, given a pair ( x, F ) ∈ ground R ( W ) × flaps R ( W ) , the set { x, v F } is an edge of W R if and only if x ∈ ∂F. We call the vertices of ground R ( W ) (resp. vflaps R ( W )) ground-vertices (resp. flap-vertices ) of W R . Notice that the incidence graph of theplane hypergraph ( N (Γ) , { ˜ c | c ∈ C (Γ) } ) is isomorphic to W R via an isomorphism that extends π and, moreover, bijectively corresponds cells to flap-vertices. This permits us to treat W R asa ∆-embedded graph where bd (∆) ∩ W R is the set X ∩ Y. As an example, see Figure 6 for the˜ R -leveling of the flat 5-wall ˜ W of Figure 3.We denote by W • the graph obtained from W if we subdivide once every edge of W that isshort in compass R ( W ) . The graph W • is a “slightly richer variant” of W that is necessary for ourdefinitions and proofs, namely to be able to associate every flap-vertex of an appropriate subgraphof W R (that we will denote by R W ) with a non-empty path of W • , as we proceed to formalize. Wesay that ( W, R ) is well-aligned if the following holds: W R contains as a subgraph an r -wall R W where D ( R W ) = D ( W R ) and W • is isomorphicto some subdivision of R W via an isomorphism that maps each ground vertex to itself.Suppose now that the flatness pair ( W, R ) is well-aligned. We call the wall R W in the abovecondition a representation of W in W R . As an example, notice that the flatness pair ( ˜ W , ˜ R ) of Figure 3 is well-aligned while the flatnesspair ( W, R ) in Figure 5 is not since, for example, in the uppermost rightmost grey cell, the upperright ground vertex can not be mapped to itself in order to yield a subgraph R W of W R as in theabove property. Lemma 16.
If a flatness pair ( W, R ) is regular, then it is also well-aligned. Moreover, there is an O ( n ) time algorithm that, given G and such a ( W, R ) , outputs a representation R W of W in W R . Proof.
Let ( W, R ) be a flatness pair where all cells of R are tidy and with no W -external or W -marginal cells. We claim that none of the cells of R is W -outer-perimetric. Indeed, a W -outer-perimetric c should correspond to one of the tree last cases of Figure 9 (this figure appears later inSubsection 4.2 in order to illustrate further definitions): in the fifth case c is untidy and in the sixthand seventh case c is W -marginal. Therefore all cells are either W -internal or W -inner-perimetricand are also all tidy.We also denote R = ( X, Y, P, C, Γ , σ, π ) . Recall that W • (whose edges are depicted in orangein Figure 7) is the graph obtained from W if we subdivide once every short edge in W. Let ξ bethe function mapping every vertex created by a subdivision of a short edge of W • (depicted by across in Figure 7) to the corresponding (trivial) flap-vertex of W R (that is depicted as one of theblue vertices of degree two).Consider R W = ( B ∪ F ∪ F , E ) , where B = W ∩ ground R ( W ) ,F = { ξ ( x ) | x is a subdivision vertex of W • } , and F = { v F ∈ vflaps R ( W ) | E ( W ∩ F ) = ∅ and F is a non-trivial flap } . In Figure 7, the vertices in B are depicted in red in Figure 7 while the vertices in F ∪ F aredepicted in blue. We define E as follows. For every v F ∈ F we include in E both edges of W R v F . For every v F ∈ F such that F \ ∂F contains a 3-branch vertex of W we includein E the three edges of W R that incident to v F . Finally, for every v F ∈ F such that F \ ∂F doesnot contain any 3-branch vertex of W we first consider the non-trivial path P F in W ∩ F and weadd in E the edges of W R between the flap-vertex v F and the endpoints of P F . Notice that since σ − ( F ) is tidy, P F does not contain internal vertices in ∂F. Observe that R W is indeed a wall of W R , where D ( W R ) = D ( R W ) , that can be computed in O ( n ) time. We now define a mapping ρ : V ( R W ) → V ( W • ) and a function τ mapping the edges in E ( R W ) (depicted as fat purple edgesin Figure 7) to subpaths of W • as follows:Figure 7: A well-aligned flatness pair ( W, R ) where W is a 3-wall, the wall W • (whose edges aredepicted in red and the new subdivision vertices are depicted by small crosses), the leveling W R of W (whose edges are depicted in purple), and the subgraph R W of W R (depicted by fat purpleedges).• If x ∈ B, then ρ ( x ) = x. • If v F ∈ F and ∂F = { x, y } , then we set ρ ( v F ) = ξ − ( v F ) , τ ( { x, v F } ) = { x, ξ − ( v F ) } , and τ ( { y, v F } ) = { y, ξ − ( v F ) } . • If v F ∈ F and v F is a branch vertex of R W , then assume first that ∂F = { x, y, z } . Because thecell σ − ( F ) is tidy the graph F \ ∂F contains a unique 3-branch vertex w of W (or equivalentlyof W • ) and F ∩ W • consists of three internally disjoint paths P w,x , P w,y , and P w,z in F from w to x, y, and z, respectively. We set ρ ( v F ) = w, τ ( { x, v F } ) = P w,x , τ ( { y, v F } ) = P w,y , and τ ( { z, v F } ) = P w,z . • If v F ∈ F and v F is not a 3-branch vertex of R W , then there exist two vertices x, y of R W such that N R W ( v F ) = { x, y } . Pick an internal vertex w of the ( x, y )-path P F and set ρ ( v F ) = w (recall that, as σ − ( F ) is tidy, none of the internal vertices of the path P F is a23round vertex). If P w,x is the ( w, x )-subpath of P F , and P w,y is the ( w, y )-subpath of P F , then set τ ( { x, v F } ) = P w,x and τ ( { y, v F } ) = P w,y . It is now easy to verify that the mappings ρ and τ defined above certify that W • is isomorphic to asubdivision of R W by an isomorphism extending ρ (see Figure 7 for an example). As all membersof B = W ∩ ground R ( W ) are, by definition, fixed points of ρ, then ( W, R ) is well-aligned. This section is devoted to the proofs of Theorem 5 and Theorem 6. We first present some definitionsin Subsection 4.1 and Subsection 4.2, necessary for the proof of the main technical lemma of thispaper, namely Lemma 17, presented in Subsection 4.3.
Let F be a graph and x and y be two distinct vertices belonging to the same connected componentof F. We say that a sequence h F , . . . , F r i of subgraphs of F is a stretching of F along the pair ( x, y ) if there is a shortest ( x, y )-path P F in F such that the sequence h F , . . . , F r i consists of the(unique) minimum-sized collection of subpaths of P F with the following properties:• each path in h F , . . . , F r i is a path where all internal vertices have degree two in F, • no two paths in h F , . . . , F r i have a common edge,• F ∪ · · · ∪ F r = P F , • for every ( i, j ) ∈ (cid:0) [ r ]2 (cid:1) , F i ∩ F j = ∅ if and only if | i − j | = 1 , and• x ∈ V ( F ) and y ∈ V ( F r ) . For an example of a streching of a graph F along a pair ( x, y ) , see Figure 8. P F F F F F F F F F \ V ( P F ) y yxx z Figure 8: The stretching of a graph F along the pair ( x, y ) . Let G be a graph and let ( W, R ) be a flatness pair of G, where R = ( X, Y, P, C, Γ , σ, π ) . Let W ∈ S R ( W ) . We now further refine the classification of the cells of R that we gave in Subsection 2.3with respect to W . See Figure 9 for an illustration of the ways a W -perimetric cell c of Γ mayintersect ∆ W . The simplest case if when | ˜ c | = 2 , depicted in the leftmost configuration of the24gure. The remaining configurations correspond to the case where ∂σ ( c ) = { x, y, z } where A c is a( π − ( x ) , π − ( y ))-arc (see Subsection 2.3 for the definition of the paths P in c and P out c , the arc A c , and the vertex z ). The second/fifth, third/sixth, and forth/seventh configurations correspond tothe case where z is an internal vertex of P in c , P out c , or none of them, respectively. This permits afurther classification of the W -perimetric cells of Γ as follows. A cell c of Γ is W -inner-perimetric (resp. W -outer-perimetric ) if c ∩ ∆ W is situated in c as indicated in the left (resp. right) part ofFigure 9. W (cid:48) -inner-perimetric cells A c W (cid:48) -outer-perimetric cells A c c ∩ ∆ W (cid:48) A c A c c ∩ ∆ W (cid:48) c ∩ ∆ W (cid:48) A c c ∩ ∆ W (cid:48) A c A c c ∩ ∆ W (cid:48) c ∩ ∆ W (cid:48) c ∩ ∆ W (cid:48) Figure 9: Seven ways ∆ W may traverse a cell. The arc A c is depicted in orange.We denote the set of cells of Γ that are W -inner-perimetric, W -outer-perimetric, W -internal,and W -strictly external by C ip W (Γ) , C op W (Γ) , C in W (Γ) , and C ex W (Γ) , respectively. See Figure 10 foran example of this further classification (relatively to Figure 5). Notice that all W -marginal cellsof Γ are W -outer-perimetric cells (corresponding to the last two cases of Figure 9). Lemma 17.
There is an algorithm that, given a graph G, a flatness pair ( W, R ) , where R =( X, Y, P, C, Γ , σ, π ) , and a wall W ∈ S R ( W ) , outputs, in O ( n + m ) time, a flatness pair ( ˜ W , ˜ R ) where ˜ R = ( X , Y , P , C , Γ , σ , π ) such that1. all cells of ˜ R are ˜ W -internal or ˜ W -inner-perimetric,2. ˜ W is a tilt of W , σ | C in ˜ W (Γ ) = σ | C in W (Γ) , i.e., the set of ˜ W -internal cells of ˜ R is the same as the set of W -internal cells of R and their images via σ and σ are also the same, and4. compass ˜ R ( ˜ W ) is a subgraph of SSSSSSSSS influence R ( W ) . Moreover, if all W -internal or W -inner-perimetric cells of R are tidy, then the flatness pair ( ˜ W , ˜ R ) is regular.Proof. Since R = ( X, Y, P, C, Γ , σ, π ) is a 7-tuple certifying that W is flat in G , we have that thetriple (Γ , σ, π ) is an Ω-rendition of G [ Y ] , where Γ = ( U, N ) is a ∆-painting.We define a series of ingredients that will permit us to define an alternative 7-tuple ˜ R . As afirst step, for every W -inner-perimetric cell c ∈ C ip W (Γ) we define an arc Y c of ∆ , as in Figure 11(where Y c is depicted in red), we set F c = σ ( c ) , r c = 1 , and V c mid = π (˜ c ) ∩ V ( D ( W )) (the verticesin V c mid are depicted in orange in Figure 11).Next, we consider a W -outer-perimetric cell c ∈ C op W (Γ) . We assume that π (˜ c ) = { x, y, z } andthat x and y are the two endpoints of the non-trivial path of D ( W ) ∩ σ ( c ) (by non-trivial we refer25igure 10: A flat wall W in a graph G, the painting of a rendition R certifying its flatness, a subwall W of W, of height three, which is R -normal, and the R -flaps of W, corresponding to the cells of R that are not W -external. The edges and the non-boundary vertices of the flaps correspondingto the W -external cells of R (depicted in pink) are not depicted (however their boundary verticesthat are not in D ( W ) are depicted in grey). There are nine W -outer-perimetric cells of R (inblue) and seven W -inner-perimetric cells (in yellow). Also, there are thirteen W -internal cells of R (in green). Among the W -inner-perimetric and W -internal cells of R , those that are untidy aredepicted with a dashed boundary. The orange cycle is the circle K W . to the path that has distinct endpoints). We also define V cW as the set of all internal endpointsof this path that are different from z. Let h F c , . . . , F cr c i be the stretching of σ ( c ) along the pair( x, y ) and let v i , for i ∈ [ r c − , be the common endpoint of F ci and F ci +1 . Notice that by tightnessproperty ( i ), r c ≥ . This permits us to set up a special vertex v c = v . We also set V c mid = { x, v , . . . , v r c − , y } , V c in = [[[[[[[[[ { V ( F ci ) | i ∈ [ r c ] } \ V c mid . Let p = π − ( x ) , p r c = π − ( y ) , and create a collection c , . . . , c r c of open disks in c and a set p , . . . , p r c − of points in c such that• p ∈ bd ( c ) and p r c ∈ bd ( c r c ) , p = p , and p r c = p r c − , • for i ∈ [ r c − , ¯ c i ∩ ¯ c i +1 = { p i } , and 26 ∩ ∆ W (cid:48) c ∩ ∆ W (cid:48) c ∩ ∆ W (cid:48) c ∩ ∆ W (cid:48) c ∩ ∆ W (cid:48) Figure 11: The four cases of the definition of the arc Y c (depicted in red), for W -inner-perimetriccells. The boundary of ∆ W is depicted in orange and the boundary of ∆ is depicted in purple.• for every ( i, j ) ∈ (cid:0) [ r c ]2 (cid:1) , ¯ c i ∩ ¯ c j = ∅ if and only if | i − j | = 1 . We define the cell replacement of c as the set c - repl ( c ) = { c , . . . , c r c } , the point replacement of c as the set p - repl ( c ) = { p , . . . , p r c } , and we set C c new = SSSSSSSSS c - repl ( c ) and N c new = SSSSSSSSS p - repl ( c ) . We also define the arc Y c as an arc of c where p i ∈ Y c , i ∈ [0 , r c ] , such that p , p r c are theextreme points of Y c , and Y c is traversing ˜ c as depicted by the red line in Figure 12. Observe that SSSSSSSSS { Y c | c ∈ C ip W (Γ) ∪ C op W (Γ) } is a “red” cycle of ∆ . Let ∆ be the disk bounded by this cycle forwhich ∆ ⊆ ∆ . c ∩ ∆ W (cid:48) c ∩ ∆ W (cid:48) c ∩ ∆ W (cid:48) c c p r c − p r c p p p Y c c r c Figure 12: The definition of the replacement sequence c , . . . , c r c and the arc Y c for the three casesof W -external cells of C op W (Γ) . We set H = [[[[[[[[[ { F c ∪ · · · ∪ F cr c | c ∈ C op W (Γ) } , V W = [[[[[[[[[ { V cW | c ∈ C op W (Γ) } ,V mid = [[[[[[[[[ { V c mid | c ∈ C ip W (Γ) ∪ C op W (Γ) } , V in = [[[[[[[[[ { V c in | c ∈ C op W (Γ) } ,N new = [[[[[[[[[ { N c new | c ∈ C op W (Γ) } , U new = [[[[[[[[[ { C c new ∪ N c new | c ∈ C op W (Γ) } . We now define the wall ˜ W = ( W \ V W ) ∪ H, i.e., we extract from W the internal vertices of thesubpaths of W that are intersected by images, via σ, of W -outer-perimetric cells and we substitutethem by the paths of their stretchings. Clearly this does not affect the interior of W , and therefore˜ W is a tilt of W , yielding Property of the statement of the lemma. Next we define a separation( X , Y ) of G so that Y = [[[[[[[[[ { V ( σ ( c )) | c ∈ C ip W (Γ) ∪ C in W (Γ) } ∪ V in ∪ V mid , X = ( V ( G ) \ Y ) ∪ V mid .
27n other words, Y consists of the images of the internal cells and the vertices of every path F ci , while X consists of everything else, except from V mid (that is, the set X ∩ Y ). Notice that G [ Y ] is a subgraph of [[[[[[[[[ { σ ( c ) | c ∈ C in W (Γ) ∪ C ip W (Γ) ∪ C op W (Γ) } = influence R ( W ) . (1)We define the pair ( P , C ) as follows. Let c be a W -outer-perimetric cell and σ ( c ) ∩ V ( D ( W ))contain a vertex w such that either w is a 3-branch vertex of W or w ∈ P (resp. w ∈ C ). Wedistinguish two cases. If w ∈ Y , then we include w in P (resp. C ). If w Y , then we includethe special vertex v c in P (resp. C ).We next define an Ω -rendition (Γ , σ , π ) of G [ Y ] where Γ = ( U , N ) is a ∆ -painting. Forthis we set Γ = ( U , N ) , where U = (cid:0)(cid:0) U \ [[[[[[[[[ C op W (Γ) (cid:1) ∩ ∆ (cid:1) ∪ U new and N = ( N ∩ ∆ ) ∪ N new . Let now K be the set of the connected components of U \ N , which will form the cells of the newΩ -rendition (Γ , σ , π ) . We define the function σ mapping the cells in C to subgraphs of G [ Y ]as follows. Notice that c ∈ K ∩ C (Γ) if and only if c ∈ C in W (Γ) ∩ C ip W (Γ) , and in this case weset σ ( c ) = σ ( c ) . Suppose now that c ∈ K \ C (Γ) . Then c should be one of the cells, say c i , of c - repl ( c ∗ ) = { c , . . . , c r c } for some c ∗ ∈ C op W (Γ) , and in this case we set σ ( c ) = F c ∗ i . It now remainsto define π : N → Y . Similarly to the definition of σ , we consider a p ∈ N and if p ∈ N ∩ N we set π ( p ) = π ( p ) . Suppose now that p ∈ N \ N. Then p should be one of the points, say p i , of p - repl ( c ∗ ) = { p , . . . , p r c } for some c ∗ ∈ C op W (Γ) and such that i ∈ [ r c ∗ − . In this case we define π ( p ) to be the unique common vertex of F c ∗ i and F c ∗ i +1 . It is now easy to verify that (Γ , σ , π ) is atight Ω -rendition of G [ Y ] and that the 7-tuple ˜ R := ( X , Y , P , C , Γ , σ , π ) certifies that ˜ W isflat in G (see Figure 13). Moreover K = C (Γ ) . Recall now that all the cells in C (Γ ) ∩ C (Γ) are either ˜ W -inner-perimetric or ˜ W -internal.Moreover, all the cells in C (Γ ) \ C (Γ) are cells as in the left part of Figure 9, therefore theyare ˜ W -inner-perimetric. This yields Property in the statement of the lemma. Notice also thatProperty follows directly from the definition of σ , as it concerns the W -internal cells of R , andthese cells are the same as the ˜ W -internal cells of ˜ R . Finally, recall that compass ˜ R ( ˜ W ) = G [ Y ]and Property follows because of (1).On the other hand, notice that all ˜ W -internal cells of ˜ R are also W -internal cells of R . More-over, if a ˜ W -inner-perimetric cell c of ˜ R is a cell of R , then c is either an W -inner-perimetric oran W -internal cell of R . On the other hand, all ˜ W -inner perimetric cells of ˜ R that are not cellsof R are cells as in the left part of Figure 9, therefore they are ˜ W -inner-perimetric and tidy. Weconclude that if all W -internal or W -inner-perimetric cells of R are tidy, then all cells of ˜ R aretidy as well. As ˜ R does not have any ˜ W -outer-perimetric cells it also does not have ˜ W -marginalcells. These two facts along with the fact that ˜ R does not have any ˜ W -external cells imply thatthe flatness pair ( ˜ W , ˜ R ) is regular.The running time follows from the fact that the substitution of W -outer-perimetric cells isbased on the stretching operation on the corresponding flaps, and this requires the computation ofshortest paths that, in total, takes O ( n + m ) time. Lemma 18.
There is an algorithm that, given a graph G and a flatness pair ( W, R ) , outputs, in O ( n + m ) time, a flatness pair ( W ? , R ? ) of G with the same height as ( W, R ) , with R ? = R , andsuch that all the W ? -internal or W ? -inner-perimetric cells of R ? are tidy. W , ˜ R ) created in the proof of Lemma 17. The wall ˜ W is the tilt of W where the updated part of ˜ W correspond to the red paths in Figure 10 whose edges are drawnin the orange cells. Proof.
Given a wall W and an R = ( X, Y, P, C, Γ , σ, π ) as above, we denote by C utd W (Γ) the set of allthe W -internal or W -inner-perimetric cells of Γ that are untidy. Notice that for every c ∈ C utd W (Γ) , | π (˜ c ) | = 3 . In what follows, we explain how to update W, while leaving ( X, Y, P, C, Γ , σ, π ) intact,in order to reduce | C utd W (Γ) | by one. Repeating this procedure clearly yields the statement claimedin the lemma.Let c ∈ C utd W (Γ) . We assume that π (˜ c ) = { x, y, z } and that z ∈ π (˜ c ) ∩ V ( W ) is a vertex of W such that two of the edges of W incident to z are edges of σ ( c ) . This implies that ¯ P = W ∩ σ ( c )is an ( x, y )-path containing z as an internal vertex. Moreover, none of the internal vertices of ¯ P , except from z, is a 3-branch vertex of W. By tightness properties (i), (ii), and (iii), there is a vertex w ∈ σ ( c ) \ π (˜ c ) and three internally vertex-disjoint paths P x , P y , and P z in σ ( c ) such that P x is a( w, x )-path, P y is a ( w, y )-path, and P z is a ( w, z )-path. If z is a 3-branch vertex of W we update W := ( W \ V ( ¯ P \{ x, y, z } )) ∪ P x ∪ P y ∪ P z (see bottom yellow cell with dashed boundary in Figure 14for an example), while, if not, we update W := ( W \ V ( ¯ P \ { x, y } )) ∪ P x ∪ P y (see the leftmost greencell with dashed boundary in Figure 14 for an example) and observe that W is again a flat wall of G, certified by ( X, Y, P, C, Γ , σ, π ) . Moreover, in the first case, z is not anymore a 3-branch vertexof W and is incident to only one edge of σ ( c ) ∩ W, while, in the second case, z is not anymore avertex of W. This implies that c is not anymore untidy and | C utd W (Γ) | is indeed reduced by one (seeFigure 14 for an example). As for each cell c that we modify we need to identify the paths P x , P y , and P z in σ ( c ) , the construction of W takes, in total, O ( n + m ) time. We finally have all the ingredients to prove our two main results.29igure 14: An illustration of the proof of Lemma 18, based on the flatness pair of Figure 13.The new flatness pair is ( W ? , R ? ) where W ? is depicted in red and R ? = R . Proof of Theorem 5.
Let ( W, R ) be a flatness pair of a graph G, where R = ( X, Y, P, C, Γ , σ, π )and W ∈ S R ( W ) . We call the algorithm of Lemma 17 on G, ( W, R ) , and W , which outputs, in O ( n + m ) time, a flatness pair ( ˜ W , ˜ R ) where ˜ R = ( X , Y , P , C , Γ , σ , π ) such that all cells of˜ R are ˜ W -internal or ˜ W -inner-perimetric (hence ˜ R does not have ˜ W -external cells), ˜ W is a tiltof W , the set of ˜ W -internal cells of ˜ R is the same as the set of W -internal cells of R and theirimages via σ and σ are also the same, and compass ˜ R ( ˜ W ) is a subgraph of SSSSSSSSS influence R ( W ) . Weobserve that ( ˜ W , ˜ R ) is a W -tilt of ( W, R ) and thus we return ( ˜ W , ˜ R ) . Notice that in the casewhere ( W, R ) is regular, all cells of R are tidy. Thus, by Lemma 17, ( ˜ W , ˜ R ) is also regular. Proof of Theorem 6.
Given a flatness pair ( W, R ) of a graph G, we first apply Lemma 18 to ( W, R )and obtain in time O ( n + m ) a flatness pair ( ˆ W ? , ˆ R ? ) of G with the same height as ( W, R ) , withˆ R ? = R , and such that all ˆ W ? -internal or ˆ W ? -inner-perimetric cells of ˆ R ? are tidy.We now apply Lemma 17 with input G, ( ˆ W ? , ˆ R ? ) , and ˆ W ? and obtain, in O ( n + m ) time, aflatness pair ( W ? , R ? ) of G such that, if ˆ R ? = ( ˆ X, ˆ Y , ˆ P , ˆ C, ˆΓ , ˆ σ, ˆ π ) and R ? = ( X, Y, P, C, Γ , σ, π ) , we have that all cells of R ? are W ? -internal or W ? -inner-perimetric (hence R ? does not have W ? -external cells), W ? is a tilt of ˆ W ? , the set of W ? -internal cells of ˆ W ? is the same as the set ofˆ W ? -internal cells of ˆ R ? and their images via σ and ˆ σ are also the same, and compass R ? ( W ? ) is asubgraph of SSSSSSSSS influence ˆ R ? ( ˆ W ? ) . Moreover, since all the ˆ W ? -internal or ˆ W ? -inner-perimetric cells ofˆ R ? are tidy, Lemma 17 implies that all ( W ? -internal or W ? -inner-perimetric) cells of R ? are tidy.Also, since none of the cells of R ? is W ? -outer-perimetric, none of the cells of R ? is W ? -marginal.These two facts together with the fact that none of the cells of R ? is W ? -external imply that( W ? , R ? ) is a regular flatness pair of G with the same height as ( W, R ) , as required.We now prove that compass R ? ( W ? ) ⊆ compass R ( W ) . First, keep in mind that compass R ? ( W ? ) ⊆ SSSSSSSSS influence ˆ R ? ( ˆ W ? ) . We observe that
SSSSSSSSS influence ˆ R ? ( ˆ W ? ) ⊆ compass ˆ R ? ( ˆ W ? ) and, since ˆ R ? = R , ompass ˆ R ? ( ˆ W ? ) = compass R ( W ) . Therefore, compass R ? ( W ? ) ⊆ compass R ( W ) . Finally, the claimed running time follows from Lemma 17 and Lemma 18.
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