Bounded-Treewidth Graphs: Chordality Is the Key to Single-Exponential Parameterized Algorithms ∗†

Bounded-Treewidth Graphs: Chordality Is the Key to Single-Exponential Parameterized Algorithms ^∗†

Édouard Bonnet

, Nick Brettell

, O-joung Kwon

, and Dániel Marx

1 Department of Computer Science, Middlesex University, London, UK 2 School of Mathematics and Statistics, Victoria University of Wellington,

Wellington, New Zealand

3 Logic and Semantics, Technische Universität Berlin, Berlin, Germany

4 Institute for Computer Science and Control, Hungarian Academy of Sciences, (MTA SZTAKI), Budapest, Hungary

Abstract

It has long been known thatFeedback Vertex Setcan be solved in time 2^O(wlogw)n^O(1)on graphs of treewidthw, but it was only recently that this running time was improved to 2^O(w)n^O(1), that is, to single-exponential parameterized by treewidth. We investigate which generalizations of Feedback Vertex Set can be solved in a similar running time. Formally, for a class of graphs P, Bounded P-Block Vertex Deletion asks, given a graph G on n vertices and positive integersk andd, whetherGcontains a setSof at mostk vertices such that each block ofG−Shas at mostdvertices and is inP. Assuming thatP is recognizable in polynomial time and satisfies a certain natural hereditary condition, we give a sharp characterization of when single-exponential parameterized algorithms are possible for fixed values ofd:

ifP consists only of chordal graphs, then the problem can be solved in time 2^O(wd2)n^O(1), ifP contains a graph with an induced cycle of length`>4, then the problem is not solvable in time 2<sup>o(wlogw)</sup>n<sup>O(1)</sup>even for fixedd=`, unless the ETH fails.

We also study a similar problem, calledBounded P-Component Vertex Deletion, where the target graphs have connected components of small size instead of having blocks of small size, and present analogous results.

1998 ACM Subject Classification G.2.1 Combinatorial Algorithms, G.2.2 Graph Algorithms

Keywords and phrases fixed-parameter tractable algorithms, treewidth, feedback vertex set

Digital Object Identifier 10.4230/LIPIcs.IPEC.2017.7

1 Introduction

Treewidth is a measure of how well a graph accommodates a decomposition into a tree-like structure. In the field of parameterized complexity, many NP-hard problems have been shown to have FPT algorithms when parameterized by treewidth; for example,Coloring,Vertex Cover,Feedback Vertex Set, andSteiner Tree. In fact, Courcelle [6] established a

∗ All authors were supported by ERC Starting Grant PARAMTIGHT (No. 280152) and ERC Consolidator Grant SYSTEMATICGRAPH (No. 725978). The third author was also supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (ERC consolidator grant DISTRUCT, agreement No. 648527).

† The full version can be found in [5],https://arxiv.org/abs/1704.06757.

© Édouard Bonnet, Nick Brettell, O-joung Kwon, and Dániel Marx;

licensed under Creative Commons License CC-BY

meta-theorem that every problem definable in MSO2 logic can be solved in linear time on graphs of bounded treewidth. While Courcelle’s Theorem is a very general tool for obtaining algorithmic results, for specific problems dynamic programming techniques usually give algorithms where the running timef(w)n^O(1) has better dependence on treewidthw. There is some evidence that careful implementation of dynamic programming (plus maybe some additional ideas) gives optimal dependence for some problems (see, e.g., [12]).

ForFeedback Vertex Set, standard dynamic programming techniques give 2^O(wlogw) n^O(1)-time algorithms and it was considered plausible that this could be the best possible running time. Hence it was a remarkable surprise when it turned out that 2^O(w)n^O(1) algorithms are also possible for this problem by various techniques: Cygan et al. [7] obtained a 3^wn^O(1)-time randomized algorithm by using the so-called Cut & Count technique, and Bodlaender et al. [2] showed there is a deterministic 2^O(w)n^O(1)-time algorithm by using a rank-based approach and the concept of representative sets. This was also later shown in the more general setting of representative sets in matroids by Fomin et al. [11].

Generalized feedback vertex set problems. We explore the extent to which these results apply for generalizations ofFeedback Vertex Set. TheFeedback Vertex Setproblem asks for a setS of at mostkvertices such thatG−Sis acyclic, or in other words, every block ofG−Sis a single edge or vertex. We consider generalizations where we allow the blocks to be some other type of small graph, such as triangles, small cycles, or small cliques; these generalizations were first studied in [4]. The main result of this paper is that the existence of single-exponential algorithms is closely linked to whether the small graphs we are allowing are all chordal or not. Formally, we consider the following problem:

BoundedP-Block Vertex Deletion Parameter: d,w Input: A graphGof treewidth at mostw, and positive integersdandk.

Question: Is there a setSof at mostkvertices inGsuch that each block ofG−S has at most dvertices and is inP?

The result of Bodlaender et al. [2] implies that whend= 2,BoundedP-Block Vertex Deletioncan be solved in time 2^O(w)n^O(1). Our main question is for which graph classesP can this problem be solved in time 2^O(w)n^O(1), when we regard das a fixed constant. A graph ischordal if it has no induced cycles of length at least 4. We show that ifP consists of only chordal graphs, then we can solve this problem in single-exponential time for fixedd.

ITheorem 1. LetP be a class of graphs that is block-hereditary, recognizable in polynomial time, and consists of only chordal graphs. Then Bounded P-Block Vertex Deletion can be solved in time 2^O(wd2)k²n on graphs withn vertices and treewidthw.

The condition thatP is block-hereditary ensures that the class of graphs with blocks inP is hereditary; a formal definition is given in Section 2. We complement this result by showing that ifP contains a graph that is not chordal, then single-exponential algorithms are not possible (assuming ETH), even for fixedd. Note that ifP is block-hereditary and contains a graph that is not chordal, then this graph contains a chordless cycle on`>4 vertices and consequently the cycle graph on `vertices is also inP.

ITheorem 2. IfP contains the cycle graph on `>4vertices, then BoundedP-Block Vertex Deletionis not solvable in time 2<sup>o(wlogw)</sup>n<sup>O(1)</sup> on graphs of treewidth at mostw even for fixed d=`, unless the ETH fails.

Baste et al. [1] recently studied the complexity of a similar problem, where the task is to find a set of vertices whose deletion results in a graph with no minor in a given collection

of graphsF, parameterized by treewidth. WhenF={C4}, this is equivalent toBounded P-Block Vertex DeletionwhereP ={K2, K3}, and the complexity they obtain in this case is consistent with our result.

Whether this lower bound of Theorem 2 is best possible when P contains a cycle on

`>4 vertices remains open. However, as partial evidence towards this, we note that when P contains all graphs, the result by Baste et al. [1] implies that thatBounded P-Block Vertex Deletion can be solved in time 2^O(wlogw)n^O(1) when d is fixed, as the minor obstruction setF consists of all of 2-connected graphs withd+ 1 vertices.

Bounded-size components. Using a similar technique, we can obtain analogous results for a slightly simpler problem, that we callBounded P-Component Vertex Deletion, where we want to remove at most kvertices such that each connected component of the resulting graph has at mostdvertices and belongs toP. If we have only the size constraint (i.e., P contains every graph), then this problem is known asComponent Order Connec- tivity[9]. Drange et al. [9] studied the parameterized complexity of a weighted variant of theComponent Order Connectivityproblem; their results imply, in particular, that Component Order Connectivity can be solved in time 2^O(klogd)n, but isW[1]-hard parameterized by onlyk ord. The corresponding edge-deletion problem, parameterized by treewidth, was studied by Enright and Meeks [10].

ITheorem 3. Let P be a class of graphs that is hereditary, recognizable in polynomial time, and consists of only chordal graphs. Then BoundedP-Component Vertex Deletion can be solved in time 2^O(wd2)k²non graphs with nvertices and treewidth w.

ITheorem 4. IfP contains the cycle graph on`>4vertices, thenBoundedP-Component Vertex Deletion is not solvable in time2<sup>o(wlogw)</sup>n<sup>O(1)</sup> on graphs of treewidth at mostw even for fixedd=`, unless the ETH fails.

The result of Baste et al. [1] implies that when P contains all graphs, Bounded P- Component Vertex Deletion can be solved in time 2^O(wlogw)n^O(1). When dis not fixed, one might ask whether Bounded P-Component Vertex Deletion admits an f(w)n^O(1)-time algorithm; that is, an FPT algorithm parameterized only by treewidth. We provide a negative answer: the problem is W[1]-hard whenP contains all chordal graphs, even parameterized by both treewidth andk. Furthermore, two stronger lower bound results hold, under the assumption of the ETH.

ITheorem 5. LetP be a hereditary class containing all chordal graphs. Then Bounded P-Component Vertex DeletionisW[1]-hard parameterized by the combined parameter (w, k). Moreover, unless the ETH fails, (1) this problem has no f(w)n^o(w)-time algorithm;

and (2) it has nof(k⁰)n^o(k0/logk0)-time algorithm, wherek⁰ =w+k.

Techniques. A pair (G, S) consisting of a graph G and a vertex subset S of G will be called a boundaried graph, and an S-block of Gis a block of Gcontaining an edge with both endpoints in S. The algorithm for Bounded P-Block Vertex Deletion uses several lemmas on S-blocks of boundaried graphs (G, S), which appear in Section 3. The key property is the following: (*) when we merge two boundaried graphs (G, S) and (H, S) into a graph G⁰, to decide whether eachS-block of G⁰ is some fixed target graph that is chordal, it is sufficient to know, for each non-trivial blockB of G[S] or H[S], some local information aboutB in theS-block containingB inGorH, respectively. We think of target graphs as labeled graphs where any two vertices in the same block have distinct labels in

{1, . . . , d}, and the local information referred to in (*) is the set of labels of neighbors ofB in theS-block containingB. The related result is stated as Proposition 6. This will be used to determine whether each of theS-blocks ofG⁰ is one of the target graphs inP. After then, to decide whetherG⁰ is a required graph, it remains to check that the whole graph has no chordless cycle, since there is a possibility of linking two controlled blocks by a sequence of uncontrolled blocks in both sidesGandH, and thus creating a chordless cycle in G⁰. This second part can be dealt with in a similar manner to the single-exponential time algorithm forFeedback Vertex Set, using representative-set techniques.

2 Preliminaries

We follow the terminology of Diestel [8], unless otherwise specified. A vertexv ofGis acut vertexif the deletion ofvfromGincreases the number of connected components. We sayG isbiconnected if it is connected and has no cut vertices. Note that every connected graph on at most two vertices is biconnected. Ablock of Gis a maximal biconnected subgraph ofG.

We sayGis 2-connected if it is biconnected and|V(G)|>3. An induced cycle of length at least four is called achordless cycle. A graph ischordal if it has no chordless cycles. For a class of graphsP, a graph is called aP-block graph if each of its blocks is inP. A classC of graphs isblock-hereditary if for every G∈ C and every biconnected induced subgraphH of G, H ∈ C. For two integers d₁, d₂ withd₁ 6d₂, let [d₁, d₂] be the set of all integers i withd16i6d2, and for a positive integer, let [d] := [1, d]. For a functionf :X→Y and X⁰⊆X, the functionf⁰:X⁰→Y where f⁰(x) =f(x) for allx∈X⁰ is called therestriction off onX⁰, and is denotedf|X⁰. We also say thatf extendsf⁰ to the set X.

Blockd-labeling. Ablockd-labelingof a graphGis a functionL:V(G)→[d] such that for each blockB ofG,L|_V(B) is an injection. IfGis equipped with a blockd-labelingL, then it is called a(block)d-labeled graph, and we callL(v) thelabel ofv. Twod-labeled graphs GandH are label-isomorphic if there is a graph isomorphism from Gto H that is label preserving. For biconnected blockd-labeled graphsGandH,H ispartially label-isomorphic toGifH is label-isomorphic to the subgraph ofGinduced by the vertices with labels inH.

Treewidth. A tree decompositionof a graphGis a pair (T,B) consisting of a treeT and a family B = {Bt}_t∈V(T) of sets Bt ⊆ V(G), called bags, satisfying the following three conditions: (1)V(G) =S

t∈V(T)B_t, (2) for every edgeuvof G, there exists a nodet ofT such thatu, v ∈Bt, (3) fort1, t2, t3∈V(T),Bt₁∩Bt₃⊆Bt₂ whenevert2is on the path from t₁ tot₃ inT. The width of a tree decomposition (T,B) is max{|B_t| −1 :t∈V(T)}. The treewidth ofGis the minimum width over all tree decompositions ofG. A tree decomposition (T,B={Bt}_t∈V(T)) isnice ifT is a rooted tree with root noder, and every nodet ofT is one of the following: (1) aleaf node: t is a leaf ofT andBt=∅; (2) anintroduce node: t has exactly one childt⁰ andBt=Bt⁰∪ {v}for somev∈V(G)\Bt⁰; (3) aforget node: t has exactly one childt⁰ andB_t=B_t⁰\ {v} for somev∈B_t⁰; or (4) ajoin node: t has exactly two childrent1 andt2, andBt=Bt₁=Bt₂.

Boundaried graphs. For a graphGandS⊆V(G), the pair (G, S) is aboundaried graph.

WhenGis ad-labeled graph, we simply say that (G, S) is ad-labeled graph. Twod-labeled graphs (G, S) and (H, S) are said to becompatibleifV(G−S)∩V(H−S) =∅,G[S] =H[S], andGandH have the same labels onS. For two compatibled-labeled graphs (G, S) and (H, S), thesumof two graphs (G, S)⊕(H, S) is the graph obtained from the disjoint union of

GandH by identifying each vertex inS and removing an edge if multiple edges appear. We denote byLG⊕LH the function fromV((G, S)⊕(H, S)) to [d] where forv∈V(G)∪V(H), (L_G⊕L_H)(v) =L_G(v) ifv∈V(G) and (L_G⊕L_H)(v) =L_H(v) otherwise. For two unlabeled

boundaried graphs, we define the sum in the same way, but ignoring the label condition.

A block of a graph is non-trivial if it has at least two vertices. For a boundaried graph (G, S), a blockB of Gis called anS-block if it contains an edge ofG[S]. Note that every non-trivial block ofG[S] is contained in a uniqueS-block ofGbecause two distinct blocks share at most one vertex. Let (G, S) be a boundaried graph. We defineAux(G, S) as the bipartite boundaried graph with bipartition (C1,C2) and boundary C2 such that (1)C1is the set of components ofG, andC2 is the set of components ofG[S], (2) forC₁∈ C1andC₂∈ C2, C1C2∈E(Aux(G, S)) if and only ifC2is contained inC1. When (G, S) and (H, S) are two compatibled-labeled graphs,Aux(G, S)⊕Aux(H, S) is well-defined, asGandH have the same set of components onS. For a setS and a setX of subsets ofS, letInc(S,X) be the bipartite graph on the bipartition (S,X) where forv∈S andX ∈ X,v andX are adjacent inInc(S,X) if and only ifv∈X. For a boundaried graph (G, S), whenP is the partition of the setCof components ofG[S] such that two components ofG[S] are in the same part if and only if they are in the same component ofG, we denote byInc(C,P)∼Aux(G, S).

3 Lemmas about S-blocks

We present several lemmas regarding S-blocks. For a biconnected d-labeled graph Q, a d-labeled graph (G, S) isblock-wise partially label-isomorphic toQif everyS-blockB ofGis partially label-isomorphic toQ. For two compatibled-labeled graphs (G, S) and (H, S) with labelingsLG andLH respectively, we say (G, S) and (H, S) areblock-wise Q-compatible if 1. (G, S) and (H, S) are block-wise partially label-isomorphic toQ; and

2. for every non-trivial block B of G[S], letting B1 andB2 be the S-blocks of Gand H that contain B, respectively, L_G(N_B1(V(B))\S)∩L_H(N_B2(V(B))\S) =∅, and, for

`1∈LG(NB<sub>1</sub>(V(B))\S) and`2∈LH(NB₂(V(B))\S), the vertices inQwith labels`1

and `₂are not adjacent.

We describe sufficient conditions for when, given a chordal labeled graph Q, the sum of two given labeled graphs (G, S) and (H, S), each partially label-isomorphic toQ, is also partially label-isomorphic toQ.

I Proposition 6. Let Q be a biconnected d-labeled chordal graph. Let (G, S) and (H, S) be two block-wiseQ-compatible d-labeled graphs such that Aux(G, S)⊕Aux(H, S) has no cycles. Then(G, S)⊕(H, S)is block-wise partially label-isomorphic toQ.

We use the following essential property of chordal graphs.

I Lemma 7. Let F be a connected graph and let Q be a connected chordal graph. Let µ:V(F)→V(Q)be a function such that for every induced pathp₁· · ·pm inF of length at most two,µ(p1), . . . , µ(pm)are pairwise distinct andµ(p1)· · ·µ(pm)is an induced path ofQ.

Thenµis an injection and preserves the adjacency relation.

ILemma 8. Let(G, S)and(H, S)be two compatibled-labeled graphs such thatAux(G, S)⊕

Aux(H, S)has no cycles. (1) IfF is anS-block of(G, S)⊕(H, S) anduv is an edge inF, then uv is contained in some S-block ofG or H. (2) Suppose eachS-block of G or H is chordal. IfF is anS-block of (G, S)⊕(H, S)anduvw is an induced path inF such that u andw are not contained in the sameS-block ofGorH, thenv∈S, and there is an induced path q1q2· · ·q` from u=q1 tow=q` inF−v such that eachqi is a neighbor of v.

Proof of Proposition 6. LetFbe anS-block of (G, S)⊕(H, S). LetLGandLHbe labelings ofGandH, respectively, and letL:=LG⊕LH. We may assume |V(F)|>3. By Lemma 8, every edge ofF is contained in someS-block ofGorH. Thus, foruv∈E(F), we haveL(u)6=

L(v) and the vertices with labelsL(u) andL(v) are adjacent inQ. Moreover, since (G, S) and (H, S) are block-wise partially label-isomorphic toQ, we haveL(V(F))⊆L_Q(V(Q)).

Letµ:V(F)→V(Q) such that for eachv∈V(F),L(v) =LQ(µ(v)).

To apply Lemma 7, it is sufficient to prove that if uvw is an induced path inF, then L(u) 6= L(w) and µ(u)µ(v)µ(w) is an induced path in Q. Since (G, S) and (H, S) are block-wise partially label-isomorphic toQ, if all ofu, v, ware contained in an S-block ofG orH, then it follows from the given condition. We may assumeuandware not contained in the sameS-block ofGorH. Then by (2) of Lemma 8,v ∈S, and there is an induced pathq₁q₂· · ·q_{`</sub> fromu=q<sub>1</sub> tow=q<sub>`} inF−v such that eachq_i is a neighbor ofv.

We show that for i ∈ {1, . . . , `−2}, L(qi), L(qi+1), L(qi+2) are pairwise distinct, and µ(qi)µ(qi+1)µ(qi+2) is an induced path ofQ. If all ofqi, qi+1, qi+2 are contained inGorH, then they are contained in the sameS-block asv, and the claim follows. We may assumeqi

andqi+2 are in distinct graphs ofG−S andH−S. Then theS-block containingqi, qi+1, v and theS-block containingq_i+1, q_i+2, vshare the edge q_i+1v. Since (G, S) and (H, S) are block-wiseQ-compatible,L(qi)6=L(qi+2) andµ(qi) is not adjacent to µ(qi+2) inQ.

We verify that µ(q₁)µ(q₂)· · ·µ(q<sub>`</sub>) is an induced path ofQ. Suppose this is false, and choosei1, i2∈ {1,2, . . . , `}withi2−i1>1 and minimumi2−i1such thatµ(qi₁) is adjacent toµ(qi₂) inQ. By minimality,µ(qi₁)· · ·µ(qi₂−1) andµ(qi₁+1)· · ·µ(qi₂) are induced paths and have length at least 2. Thusµ(qi1)· · ·µ(qi2) is an induced cycle of length at least 4, contradicting the assumption thatQis chordal. Therefore,µ(q1)µ(q2)· · ·µ(q`) is an induced path ofQ, and, in particular, L(u)6=L(w) andµ(u) andµ(w) are not adjacent in Q, as required. By Lemma 7, we conclude thatF is partially label-isomorphic toQ. J

Using Lemma 8, we can also prove the following.

ILemma 9. LetAbe a set, let(G, S)and(H, S)be two compatibled-labeled graphs, and let Bbe the set of non-trivial blocks inG[S]. Supposeg:B →Ais a function where eachS-block of G or H is chordal, Aux(G, S)⊕Aux(H, S) has no cycles, and for every B1, B2 ∈ B whereB1 andB2 are contained in anS-block of GorH,g(B1) =g(B2). IfF is anS-block of (G, S)⊕(H, S)andB₁, B₂∈ Bwhere V(B₁), V(B₂)⊆V(F), theng(B₁) =g(B₂).

I Proposition 10. Let (G, S) and (H, S) be two compatible d-labeled graphs such that every S-block of (G, S)⊕(H, S)is chordal. Then(G, S)⊕(H, S)is chordal if and only if Aux(G, S)⊕Aux(H, S)has no cycles.

Proof. We briefly sketch the proof of one direction. Suppose thatAux(G, S)⊕Aux(H, S) has a cycleC1−A1−C2−A2− · · · −Cn−An −C1 where C1, . . . , Cn are components of G[S]. For eachi ∈ {1, . . . , n}, let P_i be the shortest path from C_i toC_i+1 in A_i, and let vi, wi be the end vertices of Pi where vi ∈ V(Ci) and wi ∈ V(Ci+1). Let Qi be the shortest path fromwi tovi+1 inCi+1. We may assumen>3; it is easy when n= 2. Then v1P1−Q1−P2−Q2− · · · −Pn−Qnv1 is a cycle in (G, S)⊕(H, S), but is not necessarily a chordless cycle. We claim that it contains a chordless cycle. Letxbe the vertex following v₂ in P₂, and let y be the vertex preceding w_n in P_n. Take a shortest path P from x toy in the pathy−Qn−P1−Q1−x. Clearly P has length at least 2, asxand y are contained in distinct connected components ofGorH. Also, every internal vertex ofP has no neighbors in the other path of the cyclev1P1−Q1−P2−Q2− · · · −Pn−Qnv1between xandy. So, if we take a shortest pathP⁰ fromxto y along the other part of the cycle v1P1−Q1−P2−Q2− · · · −Pn−Qnv1, thenP∪P⁰ is a chordless cycle. J

4 Bounded P-Block Vertex Deletion

We prove Theorem 1. We first focus on S-blocks of boundaried graphs (G, S). For each non-trivial block of G[S], we guess its final shape, as ad-labeled biconnected graph, and store the labelings of the vertices and their neighbors in the S-block of G containing it.

Collectively, we call this information a characteristic of (G, S). Using characteristics, we controlS-blocks in (G, S)⊕(H, S), where (H, S) is a compatibled-labeled graph. By the previous step, we may assume that everyS-block of (G, S)⊕(H, S) is inP and has at most dvertices. Note that (G, S)⊕(H, S) still may have a chordless cycle. By Proposition 10, if we assume that everyS-block of (G, S)⊕(H, S) is inP, then (G, S)⊕(H, S) is chordal if and only if Aux(G, S)⊕Aux(H, S) has no cycles. So, instead of keepingAux(G, S), we store the corresponding partition of the set of components ofG[S].

For convenience, we fix an integer d>2 and a classP of graphs that is block-hereditary, recognizable in polynomial time, and consists of only chordal graphs. Let Ud be the set of all d-labeled biconnected P-block graphs, where each H inUd has labeling LH. For a boundaried graph (G, S), we denote by Block(G, S) the set of all non-trivial blocks inG[S].

For a d-labeled graph (G, S) with a labelingL, acharacteristic of (G, S) is a pair (g, h) of functions g : Block(G, S)→ U_d andh: Block(G, S)→2^[d] satisfying the following, for eachB∈Block(G, S) and the uniqueS-block H ofGcontainingB,

1. (label-isomorphic condition)H is partially label-isomorphic tog(B);

2. (coincidence condition) for every B⁰∈Block(G, S) withV(B⁰)⊆V(H),g(B⁰) =g(B);

3. (neighborhood condition) h(B) =L(NH(V(B))\S); and

4. (complete condition) for every wwherew∈V(H)\S or{w}=V(H)∩V(C) for some component C of G[S], H[NH[w]] is label-isomorphic tog(B)[N_g(B)[z]] where z is the vertex ing(B) with labelL(w).

We say that the sum (G, S)⊕(H, S)respects(g, h) if for eachB ∈Block(G, S), theS-block of (G, S)⊕(H, S) containing B is label-isomorphic to g(B). The following is the main combinatorial result regarding characteristics.

I Theorem 11. Let (G₁, S), (G₂, S), (H, S) be d-labeled P-block graphs such that each (Gi, S)is compatible with (H, S), (G1, S)and (G2, S) have the same characteristic (g, h), andAux(G2, S)⊕Aux(H, S)has no cycles. If(G1, S)⊕(H, S)is a d-labeled P-block graph that respects(g, h), then(G₂, S)⊕(H, S)is ad-labeled P-block graph that respects(g, h).

Proof. We show (G₂, S)⊕(H, S) respects (g, h). Choose a non-trivial blockB ofG₂[S], let Q:=g(B), let F be theS-block of (G2, S)⊕(H, S) containingB,LF be the function from V(F) to [d] that sends each vertex to its label fromG2 orH, andLQ be the labeling ofQ.

We claim thatF is label-isomorphic toQ. We regardF as the sum of (F∩G₂, V(F)∩S) and (F∩H, V(F)∩S) and verify the conditions of Proposition 6. Using Lemma 9, for every B⁰∈Block(G₂, S) withV(B⁰)⊆V(F),g(B⁰) =Q. We also observe thatAux(F∩G₂, S_F)⊕ Aux(F∩H, SF) has no cycles asAux(G2, S)⊕Aux(H, S) has no cycles. Since (g, h) is a characteristic of (G₂, S) and (G₁, S)⊕(H, S) respects (g, h), we can confirm that both F∩GandF∩H are block-wise partially label-isomorphic toQ. The second condition of being block-wiseQ-compatible follows from the fact that (G1, S) and (G2, S) have the same characteristic (g, h). Thus,F∩G₂ andF∩H are block-wiseQ-compatible, and this implies thatF is partially label-isomorphic toQby Proposition 6. By the ‘complete condition’ of a characteristic, we can show thatL_Q(V(Q))⊆L_F(V(F)), soF is label-isomorphic to Q.

Lastly, we can confirm that (G2, S)⊕(H, S) is ad-labeled P-block graph by showing that every non S-block of (G₂, S)⊕(H, S) is fully contained inG₂ orH. We can argue this using the fact that (G2, S)⊕(H, S) is chordal, which is implied by Proposition 10. J

Proof of Theorem 1. We obtain a nice tree decomposition (T,B={Bt}_t∈V(T)) ofGwith root noder and width at most 5w+ 4 in time O(c^w·n) for some constant c using the approximation algorithm by Bodlaender et al. [3]. Fort∈V(T), letG_tbe the subgraph of Ginduced by the union of all bags Bt⁰ wheret⁰ is a descendant oft. Let Comp(t, X) be the set of all components ofG[B_t\X], and Part(t, X) be the set of all partitions of Comp(t, X).

For each nodetofT,X ⊆Bt, and a functionL:Bt\X →[d], we defineF(t, X, L) as the set of all pairs (g, h) consisting of functionsg: Block(t, X)→ Ud andh: Block(t, X)→2^[d]. We say that (g, h) isvalid, if (1)Lis ad-labeling ofG[B_t\X], (2) for eachB ∈Block(t, X),B is partially label-isomorphic tog(B), and (3) for eachB∈Block(t, X),L(V(B))∩h(B) =∅.

For i ∈ {0,1, . . . , k} and (g, h) ∈ F(t, X, L), let c[t,(X, L, i,(g, h))] be the family of all partitionsX ∈Part(t, X) satisfying the following property: there existS⊆V(Gt)\Btwith

|S|=iand ad-labelingL⁰ofGt−(X∪S) where (1)L=L⁰|_Bt\X, (2)Gt−(X∪S) is aP-block graph, (3) (g, h) is a characteristic of (Gt−(X∪S), Bt\X), and (4)Inc(Comp(t, X),X)∼ Aux(Gt−(X∪S), Bt\X). Such a pair (S, L⁰) is apartial solution with respect to X.

The main idea is that instead of fully computing c[t, M] for M = (X, L, i,(g, h)), we recursively enumerate a setr[t, M] that may represent partial solutions forc[t, M]. Formally, for a subset r[t, M] ⊆c[t, M], we denote r[t, M] ≡c[t, M] if for every X ∈ c[t, M] and a partial solution (S, L⁰) with respect toX andSout⊆V(G)\V(Gt) whereG−(S∪X∪Sout) is ad-labeledP-block graph respecting (g, h), there existsX1∈r[t, M] and a partial solution (S⁰, L⁰⁰) with respect toX1such thatG−(S⁰∪X∪Sout) is ad-labeledP-block graph respecting (g, h). By the definition ofr[t, M], the problem is aYes-instance if and only if there exists (X, L, i,(g, h)) for the root node rwith|X|+i6ksuch thatr[r,(X, L, i,(g, h)]6=∅.

Whenever we update r[t, M], we confirm that |r[t, M]| 6w·2^w−1. This will be the application of the representative set technique developed by Bodlaender et al. [2]. For a setS and a set Aof partitions of S, a subset A⁰ of Ais called a representative set if for everyX1∈ Aand every partition Y ofS whereInc(S,X1∪ Y) has no cycles, there exists a partitionX₂∈ A⁰ such thatInc(S,X₂∪ Y) has no cycles.

IProposition 12. Given a familyAof partitions of a setS, one can output a representative set ofAof size at most |S| ·2^|S|−1 in timeA^O(1)2^O(|S|).

We sketch how to update familiesr[t, M] whentis an introduce node with child nodet⁰. We may assume (g, h) is valid, otherwisec[t, M] =∅.

Let v be the vertex in Bt \Bt⁰. If v ∈ X, then Gt−X = Gt⁰ −(X \ {v}) and B_t\X =B_t⁰\(X\ {v}). Thus, we can setr[t, M] :=r[t⁰,(X\ {v}, L, i,(g, h))]. We assume v /∈ X, and let Lres := L|B_t0\X. For (g, h) ∈ F(t, X, L), a pair (g⁰, h⁰) ∈ F(t⁰, X, Lres) is called the restriction of (g, h) if (1) for B1 ∈ Block(t⁰, X) and B2 ∈ Block(t, X) with V(B₁)⊆V(B₂),g⁰(B₁) =g(B₂), and ifv∈V(B₂), then every vertex ing⁰(B₁) with label inh⁰(B1) is not adjacent to the vertex ing⁰(B1) with labelL(v), (2) forB1∈Block(t⁰, X) andB₂ ∈ Block(t, X) with V(B₁) ⊆V(B₂) and v /∈V(B₂), h⁰(B₁) = h(B₂), and (3) for B2∈Block(t, X) containing v, h(B2) =S

B₁∈Block(t⁰,X),V(B₁)⊆V(B₂)h(B1).

IClaim 13. ForX ∈Part(t, X),X ∈c[t, M] if and only if there exist a restriction (g⁰, h⁰) of (g, h) and Y ∈ c[t⁰,(X, Lres, i,(g⁰, h⁰))] such that (1) v has neighbors on at most one component in each part ofY, and (2) ifv has at least one neighbor inG[B_t\X], thenX is the partition obtained from Y by, for partsY1, . . . , Ymof Y containing components having a neighbor ofv, removing all ofY₁, . . . , Ym and adding a part that consists of all components of G[Bt\X] not contained in parts ofY \ {Y1, . . . , Ym}; and otherwise, X =Y ∪ {{v}}.

We update r[t, M] as follows. Set K := ∅. For a pair of functions (g⁰, h⁰), we test whether (g⁰, h⁰) is a restriction of (g, h). Assume (g⁰, h⁰) is a restriction of (g, h). For each Y ∈r[t⁰,(X, L_res, i,(g⁰, h⁰))], we check the two conditions for (g⁰, h⁰) andY in Claim 13, and if they are satisfied, then add the setX described in Claim 13 toK; otherwise, skip it. The whole procedure can be done in time 2^O(wd2). After we do this for all possible candidates, we take a representative set ofK using Proposition 12, and assign the resulting set tor[t, M].

We claim thatr[t, M]≡c[t, M]. LetGout:=G−(V(Gt)\Bt),X ∈c[t, M], and (S, L⁰) be a partial solution with respect toX, and suppose there existsSout ⊆V(G)\V(Gt) where (G_t−(X∪S), B_t\X)⊕(G_out−(X∪S_out), B_t\X) is ad-labeledP-block graph respecting (g, h). Every (Bt⁰\X)-block ofG−(S∪X∪Sout) is chordal as such a block is a (Bt\X)-block ofG−(S∪X∪S_out). SinceG−(S∪X∪S_out) is chordal, by Proposition 10,Aux(G_t⁰−(X∪ S), Bt⁰\X)⊕Aux(Gout−(X∪Sout), Bt⁰\X) has no cycles. LetMres:= (X, Lres, i,(g⁰, h⁰)).

As r[t⁰, Mres]≡c[t⁰, Mres], there exist Y ∈r[t⁰, Mres] and a partial solution (S⁰, L⁰⁰) with respect toY such thatInc(Comp(t⁰, X),Y)∼Aux(Gt⁰−(X∪S⁰), Bt⁰\X) has no cycles.

By Theorem 11,G−(S⁰∪X∪Sout) is ad-labeledP-block graph respecting (g, h).

By the procedure,X1whereInc(Comp(t, X),X1)∼Aux(Gt−(X∪S⁰), Bt\X) is added toK. And there existX₂∈r[t, M] and a partial solution (S⁰⁰, L⁰⁰⁰) with respect toX₂ such thatG−(S⁰⁰∪X∪Sout) is ad-labeledP-block graph. Thus,r[t, M]≡c[t, M].

Total running time. We denote|V(G)|byn. Note that the number of nodes inT isO(wn).

For fixed t ∈ V(T), there are at most 2^w+1 possible choices for X ⊆ Bt, and for fixed X ⊆B_t, there are at mostd^w+1 possible functionsL. Furthermore, the size ofF(t, X, L) is bounded by 2^O(wd2). Thus, there areO(n·k·max(2, d)^w+1·2^O(wd2)) tables. In summary, the algorithm runs in timeO(n·k·max(2, d)^w+1)·2^O(wd2)·k= 2^O(wd2)k²n. J

5 Lower bound for fixed d

We showed thatBounded P-Component Vertex DeletionandBounded P-Block Vertex Deletionadmit single-exponential time algorithms parameterized by treewidth, wheneverP is a class of chordal graphs. We now establish that, assuming the ETH, this is no longer the case whenP contains a graph that is not chordal.

In thek×k Independent Setproblem, one is given a graphG= ([k]×[k], E) over the k² vertices of ak-by-k grid. We denote byhi, jiwithi, j∈[k] the vertex ofGin thei-throw andj-thcolumn. The goal is to find an independent set of size kinGthat contains exactly one vertex in each row. ThePermutation k×k Independent Setproblem is similar but with the additional constraint that the independent set should also contain exactly one vertex per column.

I Theorem 14. If P contains the cycle graph on ` > 4 vertices, then Bounded P- Component Vertex Deletion, or Bounded P-Block Vertex Deletion, is not solvable in time 2<sup>o(wlogw)</sup>n<sup>O(1)</sup> on graphs of treewidth at mostweven for fixedd=`, unless the ETH fails.

Proof. To prove this theorem, we reduce fromPermutationk×kIndependent Setwhich, likePermutationk×k Clique, cannot be solved in time 2^o(klogk)k^O(1) unless the ETH fails [13]. LetG= ([k]×[k], E) be an instance of Permutationk×k Independent Set. We assume that∀h, i, j∈[k] withh6=i,hi, jihh, ji ∈E. Adding these edges does not change theYes- andNo-instances, but has the virtue of makingPermutationk×kIndependent Setequivalent tok×kIndependent Set. We also assume that∀h, i, j∈[k],hi, jihi, hi∈/E,

S^e1 H^e1 S^e2 H^e2 S^e3 H^e3 S^em H^em

Figure 1A high-level schematic ofG⁰ andG⁰⁰. TheH^eis only differ by a constant number of edges (in red/light gray) that encode their edgeeiofG.

since at most one ofhi, jiandhi, hican be in a given solution. Letm:=|E|=O(k⁴) be the number of edges ofG.

Outline. We build two graphsG⁰ = (V⁰, E⁰) andG⁰⁰ = (V⁰, E⁰⁰) with treewidth at most (3d+ 4)k+ 6d−5 =O(k), and ((3d−2)k²+ 2k)mvertices, where the following are equivalent:

1. Ghas an independent set of sizek with one vertex per row ofG.

2. There is a set S ⊆ V⁰ of size at most (3d−2)k(k−1)m such that each connected component ofG⁰−S has size at mostdand belongs toP.

3. There is a setS⊆V⁰ of size at most (3d−2)k(k−1)msuch that each block ofG⁰⁰−S has size at mostdand belongs toP.

The overall construction ofG⁰ andG⁰⁰ will displaymalmost copies of the encoding of an edgeless G arranged in a cycle. Each copy embeds one distinct edge of G. The point of having the information ofGdistilled edge by edge inG⁰ andG⁰⁰ is to control the treewidth.

This general idea originates from a paper of Lokshtanov et al. [12].

Construction. We first describe G⁰. As a slight abuse of notation, a gadget (and, more generally, a subpart of the construction) may refer to either a subset of vertices or to an induced subgraph. For eache=hi^e, j^eihi^0e, j^0ei ∈E, we detail the internal construction of H^e andS^eof Figure 1 and how they are linked to one another. Each vertexv=hi, jiofG is represented by a gadgetH^e(v) on 3d−2 vertices inG⁰: a path on d−3 vertices whose endpoints arev^e_−aandv_−b^e , an isolated vertexv^e₊, and two disjoint cycles of lengthd. Observe that ifd= 4, thenv^e_−a andv_−b^e is the same vertex. We add all the edges betweenH^e(hi, ji) andH^e(hi, j⁰i) for i, j, j⁰ ∈[k] withj 6=j⁰. We also add all the edges betweenH^e(hi^e, j^ei) andH^e(hi^0e, j^0ei). We callH^e the graph induced by the union of everyH^e(v), forv∈V(G).

Therow/column selector gadgetS^econsists of a set S_r^eofkvertices with one vertex r_i^efor each row indexi∈[k], and a setS_c^eofk vertices with one vertexc^e_j for each column index j∈[k]. The gadgetS^eforms an independent set of size 2k. We arbitrarily number the edges ofG: e1, e2, . . . , em. For eachh∈[m] andv=hi, ji ∈V, we linkv_−a^eh tor^e_i^h (the row index ofv) andv_−b^eh toc^e_j^h (the column index ofv). We also link, for everyh∈[m−1],v^e₊^h tor_i^eh+1 and toc^e_j^h+1, andv₊^em tor^e_i¹ and toc^e_j¹. That concludes the construction (see Figure 2). To obtainG⁰⁰fromG⁰, we add the edgesc^e_j^hc^e_j+1^h for everyh∈[m] andj∈[k−1]. We ask for a deletion setS of size s:= (3d−2)k(k−1)m.

Treewidth ofG⁰ andG⁰⁰. For any edgee∈E, we setH(e) :=H^e(hi^e, j^ei)∪H^e(hi^0e, j^0ei).

For anyi∈[m−1], we set ˜Si:=S^e1∪S^ei∪S^ei+1, and ˜Sm:=S^e1∪S^em. For eache∈E, andi∈[k],H^e(i) denotes the union of theH^e(v) for all verticesv of thei-th row. Here is a path decomposition ofG⁰ andG⁰⁰:

S˜₁∪H(e₁)∪H^e1(1)→S˜₁∪H(e₁)∪H^e1(2)→. . .→S˜₁∪H(e₁)∪H^e1(k)→ ...

S˜m∪H(em)∪H^em(1)→S˜m∪H(em)∪H^em(2)→. . .→S˜m∪H(em)∪H^em(k).

row index column index

S^er¹ S^ec¹

H^e1

row index column index

S^e_r² S^e_c²

H^e2

...

Figure 2The overall picture ofG⁰ andG⁰⁰withk= 3. Dotted edges are subdividedd−4 times;

ifd= 4, they are simply edges. Dashed edges are subdividedd−5 times; ifd= 4, the two endpoints are in fact a single vertex. Edges between two boxes link each vertex of one box to each vertex of the other box. The gray edges in the column selectorsSc^eh are only present inG⁰⁰.

As, for anyh∈[m],|S˜h|66k,|H(eh)|= 2(3d−2), and|H^eh(i)|6(3d−2)kfor anyi∈[k], the size of a bag is bounded by maxh∈[m],i∈[k]|S˜h∪H(eh)∪H^eh(i)|66k+2(3d−2)+(3d−2)k= (3d+ 4)k+ 6d−4.

Correctness. If there is an independent set I of size k in G, a solution to a Bounded P-Component Vertex DeletionorBounded P-Block Vertex Deletioninstance can be obtained by deleting from eachH^eevery H^e(v) such thatv /∈I.

We show that 2 ⇒ 1 and 3 ⇒ 1. We assume that there is a set S ⊆ V⁰ of size at most s such that all the blocks ofG⁰⁰−S (resp. G⁰−S) have size at most d. We note that this corresponds to assuming condition 3 (resp. a weaker assumption than condition 2) holds. We show that there are at most 3d−2 vertices of H^e(i) remaining inG⁰⁰−S (or G⁰−S). Assume, for the sake of contradiction, that H^e(i)−S contains at least 3d−1 vertices. Observe that H^e(i)−S cannot contain at least one vertex from three distinct H^e(u), H^e(v), and H^e(w) (with u, v andw in the i-th row of G), since then H^e(i)−S would be 2-connected (and of size> d). For the same reason, H^e(i)−S cannot contain at least two vertices inH^e(u) and at least two vertices in anotherH^e(v). Therefore, the only way of fitting 3d−1 vertices inH^e(i)−S is the 3d−2 vertices of anH^e(u) plus one vertex from some otherH^e(v). But then, this vertex ofH^e(v) would form, together with oneCd ofH^e(u), a 2-connected subgraph ofG⁰⁰−S (orG⁰−S) of sized+ 1. Now, we know that|H^e(i)∩S|>(3d−2)(k−1). As there are preciselymksetsH^e(i) inG⁰ (and they are disjoint), it further holds that|H^e(i)∩S|= (3d−2)(k−1), since otherwiseS would contain strictly more thans= (3d−2)k(k−1)mvertices. Thus,H^e(i)−S contains exactly 3d−2 vertices. By the previous remarks,H^e(i)−S can only consist of the 3d−2 vertices of the sameH^e(u) or 3d−3 vertices ofH^e(u) plus one vertex from another H^e(v). In fact, the latter case is not possible, since the vertex ofH^e(v) would form, with at least one remaining Cd of the 3d−3 vertices ofH^e(u), a 2-connected subgraph ofG⁰⁰−S (orG⁰−S) of size d+ 1. This is why we needed two disjointCds in the construction instead of just one. So far, we have proved that, assuming condition 2 or condition 3 holds, for anye∈E andi∈[k], H^e(i)∩S=H^e(vi,e) for some vertexvi,e of thei-th row ofG, and for anye∈E,S^e∩S=∅.

In what follows, we show thatvi,e does not depend one. Formally, we want to show that there is avi such that, for anye∈E,vi,e=vi. Observe that it is enough to derive that, for anyh∈[m], v_i,eh =v_i,eh+1 (withe_m+1=e₁). Let j∈[k] (resp.j⁰∈[k]) be the column of vi,e_h (resp.vi,e_h+1) inG. We first assume condition 2 holds. For anyh∈[m], vi,e_heh

+, r_i^eh+1, c^e_j^h+10 , c^e_j^h+1 plus the path Pv^e_i,eh+1^h+1 (betweenv_i,eh+1^e_−a^h+1 andv_i,eh+1^e_−b^h+1) induces a path (in particular, a connected subgraph) of sized+ 1 inG⁰⁰−S, unless j=j⁰ (withem+1=e1).

Therefore, j =j⁰. As vi,e_h andvi,e_h+1 have the same column j and the same rowi inG, vi,e_h =vi,e_h+1. Showing the same property under 3 is done similarly. We can now safely definevi:=vi,e and conclude by proving that{v1, v2, . . . , vk}is a clique. J

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