Interval Deletion is Fixed-Parameter Tractable

Transcription

1 0 Interval Deletion i Fixed-Parameter Tractable YIXIN CAO and DÁNIEL MARX, Hungarian Academy of Science (MTA SZTAKI) We tudy the minimum interval deletion problem, which ak for the removal of a et of at mot k vertice to make a graph of n vertice into an interval graph. We preent a parameterized algorithm of runtime 10 k n O(1) for thi problem, that i, we how the problem i fixed-parameter tractable. Categorie and Subject Decriptor: G.2.2 [Graph Theory]: Graph algorithm; F.2.2 [Nonnumerical Algorithm and Problem]: Computation on dicrete tructure General Term: Algorithm Additional Key Word and Phrae: Ateroidal triple, congenial hole, modular decompoition ACM Reference Format: Yixin Cao and Dániel Marx Interval Deletion i Fixed-Parameter Tractable. ACM Tran. Algor. 0, 0, Article 0 ( 20xx), 34 page. DOI: 1. INTRODUCTION A graph i an interval graph if it vertice can be aigned to interval of the real line uch that there i an edge between two vertice if and only if their correponding interval interect. Interval graph are the natural model for DNA chain in biology and many other application, among which the mot cited one include job cheduling in indutrial engineering [Bar-Noy et al. 2001] and eriation in archeology [Kendall 1969]. Motivated by pure contemplation of combinatoric and practical problem of biology repectively, Hajó [1957] and Benzer [1959] independently initiated the tudy of interval graph. Interval graph are a proper ubet of chordal graph. After more than half century of intenive invetigation, the propertie and the recognition of interval and chordal graph are well undertood [Booth and Lueker 1976]. More generally, many NP-hard problem (coloring, maximum independent et, etc.) are known to be polynomial-time olvable when retricted to interval or chordal graph. Therefore, one would like to generalize thee reult to graph that do not belong to thee clae, but cloe to them in the ene that they have only a few erroneou / miing edge or vertice. A a firt tep in undertanding uch generalization, one would like to know how far the given graph i from the cla and to find the erroneou/miing element. Thi lead u naturally to the area of graph modification problem, where given a graph G, the tak i to apply a minimum number of operation on G to make it a member of ome precribed graph cla F. Depending on the operation we allow, we can conider, e.g., completion (edge-addition), edge-deletion, and vertex-deletion An extended abtract of the paper wa preented at the 25th Annual ACM-SIAM Sympoium on Dicrete Algorithm (SODA 2014). Thi work i upported by the European Reearch Council (ERC) grant and the Hungarian Scientific Reearch Fund (OTKA) grant NK Author addree: Y. Cao, (Current addre), Department of Computing, The Hong Kong Polytechnic Univerity, Hong Kong, China; D. Marx, Intitute for Computer Science and Control, Hungarian Academy of Science, Budapet, Hungary. Permiion to make digital or hard copie of part or all of thi work for peronal or claroom ue i granted without fee provided that copie are not made or ditributed for profit or commercial advantage and that copie how thi notice on the firt page or initial creen of a diplay along with the full citation. Copyright for component of thi work owned by other than ACM mut be honored. Abtracting with credit i permitted. To copy otherwie, to republih, to pot on erver, to reditribute to lit, or to ue any component of thi work in other work require prior pecific permiion and/or a fee. Permiion may be requeted from Publication Dept., ACM, Inc., 2 Penn Plaza, Suite 701, New York, NY USA, fax +1 (212) , or permiion@acm.org. c 20xx ACM /20xx/-ART0 $15.00 DOI:

2 0:2 Y. Cao and D. Marx verion of thee problem. Let u point out that, when F i hereditary, the vertex deletion verion can be conidered a the mot robut variant, which in ome ene encompae both edge addition and edge deletion: if G can be made a member of F by k 1 edge addition and k 2 edge deletion, then it can be alo made a member of F by deleting at mot k 1 + k 2 vertice (e.g., by deleting one endpoint of each added/deleted edge). Unfortunately, mot of thee graph modification problem are computationally hard: for example, a claical reult of Lewi and Yannakaki [1980] how that the vertex deletion problem i NP-hard for every nontrivial and hereditary cla F, and according to Lund and Yannakaki [1993], they are alo MAX SNP-hard. Therefore, early work of Kaplan et al. [1999] and Cai [1996] focued on the fixed-parameter tractability of graph modification problem. Recall that a problem, parameterized by k, i fixed-parameter tractable (FPT) if there i an algorithm with runtime f(k) n O(1), where f i a computable function depending only on k [Downey and Fellow 2013]. In the pecial cae when the deired graph cla F can be characterized by a finite number of forbidden (induced) ubgraph, then fixed-parameter tractability of uch a problem follow from a baic bounded earch tree algorithm [Cai 1996]. However, many important graph clae, uch a foret, bipartite graph, and chordal graph have minimal obtruction of arbitrarily large ize (cycle, odd cycle, and hole, repectively). It i much more challenging to obtain fixed-parameter tractability reult for uch clae, ee reult on, e.g., bipartite graph [Reed et al. 2004; Kawarabayahi and Reed 2010], planar graph [Marx and Schlotter 2012; Kawarabayahi 2009], acyclic graph [Cao et al. 2010; Chen et al. 2008], and minor-cloed clae [Adler et al. 2008; Fomin et al. 2012]. For interval graph, the fixed-parameter tractability of the completion problem wa raied a an open quetion by Kaplan et al. [1999] in 1994, to which a poitive anwer with a k 2k n O(1) -time algorithm wa given by Villanger et al. [2009] in In thi paper, we anwer the complementary quetion on vertex deletion: Theorem 1.1 (Main reult). There i a 10 k n O(1) -time algorithm for deciding whether or not there i a et of at mot k vertice whoe deletion make an n-vertex graph G an interval graph. Related work. Let u put our reult into context. Interval graph form a ubcla of chordal graph, which are graph containing no induced cycle of length greater than 3 (alo called hole). In other word, the minimal obtruction for being a chordal graph might be hole of arbitrary length, hence infinitely many of them. Even o, chordal completion (to make a graph chordal by the addition of at mot k edge) can till be olved by a bounded earch tree algorithm by oberving that a large hole immediately implie a negative anwer to the problem [Kaplan et al. 1999; Cai 1996]. No uch imple argument work for chordal deletion (to make the graph chordal by removing at mot k edge/vertice) and it fixedparameter tractability wa procured by a completely different and much more complicated approach [Marx 2010]. It i known that a graph i an interval graph if and only if it i chordal and doe not contain a tructure called ateroidal triple (AT for hort), i.e., three vertice uch that each pair of them i connected by a path avoiding neighbor of the third one [Lekkerkerker and Boland 1962]. Therefore, in the graph modification problem related to interval graph, one ha to detroy not only all hole, but all AT a well. The algorithm of Villanger et al. [2009] for the interval completion problem firt detroy all hole by the ame bounded earch tree technique a in chordal completion. Thi tep i followed by a delicate analyi of the AT and a complicated branching tep to break them in the reulting chordal graph. A ubcla of interval graph that received attention i the cla of unit interval graph: graph that can be repreented by interval of unit length. Interetingly, thi cla coincide with proper interval graph, which are thoe graph that have a repreentation with no

3 Interval Deletion i Fixed-Parameter Tractable 0:3 interval containing another one. It i known that unit interval graph can be characterized a not having hole and three other pecific forbidden ubgraph, thu graph modification problem related to unit interval graph [Kaplan et al. 1999; van t Hof and Villanger 2013] are very different from thoe related to interval graph, where the minimal obtruction include an infinite family of AT. Our technique. Even though both chordal deletion and interval completion eem related to interval deletion, our algorithm i completely different from the publihed algorithm for thee two problem. The algorithm of Marx [2010] for chordal deletion i baed on iterative compreion, identifying irrelevant vertice in large clique, and the ue of Courcelle Theorem on a bounded treewidth graph; none of thee technique appear in the preent paper. Villanger et al. [2009] ued a imple bounded earch tree algorithm to try every minimal way of completing all the hole; therefore, one can aume that the input graph i chordal. AT in a chordal graph are known to have the property of being hallow, and in a minimal witne of an AT, every vertex of the triple i implicial. Thi mean that the algorithm of [Villanger et al. 2009] can focu on completing uch AT. On the other hand, there i no imilar upper bound known on the number of minimal way of breaking all hole by removing vertice, and it i unlikely to exit. Therefore, in a ene, interval deletion i inherently harder than interval completion: in the former problem, we have to deal with two type of forbidden tructure, hole and hallow AT, while in the econd problem, only hallow AT concern u. Indeed, we pend ignificant effort in the preent paper to make the graph chordal; the main part of the proof i undertanding how hole interact and what the minimal way of breaking them are. The main technical idea to handle hole i developing a reduction rule baed on the modular decompoition of the graph and analyzing the tructural propertie of reduced graph. It turn out that the hole remaining in a reduced graph interact in a very pecial way (each hole i fully contained in the cloed neighborhood of any other hole). Thi property allow u to prove that the number of minimal way of breaking the hole i polynomially bounded, and thu a imple branching tep can reduce the problem to the cae when the graph i chordal. A another conequence of our reduction rule, we can prove that thi chordal graph already ha a tructure cloe to interval graph (it ha a clique tree that i a caterpillar). We can how that in uch a chordal graph, AT interact in a well-behaved way and we can find a et of 10 vertice uch that there alway exit a minimum olution that contain at leat one of thee 10 vertice. Therefore, we can complete our algorithm by branching on the deletion of one of thee vertice. Motivation. The motivation for the graph modification problem tudied in thi paper i twofold: theoretical and coming from application. Many claical graph-theoretic problem can be formulated a graph deletion to pecial graph clae. For intance, vertex cover, feedback vertex et, cluter vertex deletion, and odd cycle tranveral can be viewed a vertex deletion problem where the cla F i the cla of all empty graph, foret, cluter graph (i.e., dijoint union of clique), and bipartite graph, repectively. Thu, the tudy of graph modification problem related to important graph clae can be een a a natural extenion of the tudy of claical combinatorial problem. In light of the importance of interval graph, it i not urpriing that there are natural combinatorial problem that can be formulated a, or computationally reduced to interval deletion, and then our algorithm for interval deletion can be applied. For intance, Narayanawamy and Subahini [2013] recently ued Theorem 1.1 a a ubroutine to olve the maximum conecutive one ub-matrix problem and the minimum convex bipartite deletion problem. A a hitorical coincidence, interval graph modification problem are motivated not only from the aforementioned theoretical tudie, but becaue they have wide application. One central problem in molecular biology i to recontruct the relative poition of clone along

4 0:4 Y. Cao and D. Marx the target DNA baed on their pairwie overlap information obtained via experimental method. Thee data are naturally formulated a a graph, where each clone i a vertex, and two clone are adjacent iff they overlap. The graph hould be an interval graph provided the relation are perfect, and the problem i then equivalent to the contruction of it interval model, which can be done in linear time. However, real data are alway inconitent and contaminated by a few but crucial error, which have to be detected and fixed. In particular, on the detection of fale-poitive error that correpond to fale edge, Goldberg et al. [1995] propoed the interval edge deletion problem (to make the graph an interval graph by the deletion of at mot k edge) and howed it NP-hardne. Thi problem i equivalent to the maximum panning interval ubgraph, and i not known to be FPT or not. Moreover, fale-negative error are alo poible, which ignificantly complicate the ituation. In thi regard, we turn to the clone (vertice) involved in erroneou relation (edge) intead of the relation themelve, and try to identify them baed on a imilar aumption. More pecifically, we tudy the interval (vertex) deletion problem, which i equivalent to finding the maximum induced interval ubgraph. Conceptually, thi formulation i capable of dealing with both fale-negative and fale-poitive. Computationally, the number of clone involved in mi-oberved relation i never larger, and believed to be ignificantly maller, than the number of erroneou relation. It might thu provide better aitance to biologit by revealing more meaningful information in le time, a proclaimed by Karp [1993]: Thu, optimization method hould be viewed not a vehicle for olving a problem, but for propoing a plauible hypothei to be confirmed or diconfirmed by further experiment. The earch for the correct olution of a recontruction problem mut inevitably be an iterative proce involving a cloe interaction between experimentation and computation. In a eriation problem of archeology, overlap information of a collection of artifact i given, and we are aked to put them in chronological order. Again we cannot expect the data to be conitent and have to deal with error firt. In particular, the famou Berge mytery tory [Golumbic 2004] i eentially a eriation problem with fale overlap information given by a cheater, and can be viewed a interval deletion with k = OUTLINE The purpoe of thi ection i to decribe the main tep of our algorithm at a high level. We ay that a et Q V (G) i an interval deletion et to a graph G if G Q i an interval graph. An interval deletion et Q i minimum if there i no interval deletion et trictly maller than Q, and it i minimal if no proper ubet Q Q i an interval deletion et. A et X of vertice i called a minimal forbidden et if X doe not induce an interval graph but every proper ubet X X doe; the ubgraph G[X] i called a minimal forbidden induced ubgraph. Clearly, et Q i an interval deletion et if and only if it interect every minimal forbidden et. Our goal i to find an interval deletion et of ize at mot k. For technical reaon, it will be convenient to define the problem a follow: interval deletion: Given a graph G and an integer parameter k, return if an interval deletion et of ize k exit, a minimum interval deletion et Q V (G); if no interval deletion et of ize k exit, NO. PHASE 1: Preproceing. The firt phae of the algorithm applie two reduction rule exhautively. They either implify the intance or branch into a contant number of intance

5 Interval Deletion i Fixed-Parameter Tractable 0:5 with trictly maller parameter value. The firt reduction rule i traightforward: we detroy every forbidden et of ize at mot 10. Reduction 1. [Small forbidden et] Given an intance (G, k) and a minimal forbidden et X of no more than 10 vertice, we branch into X intance, (G v, k 1) for each v X. A graph on which Reduction 1 cannot be applied i called prereduced. The econd reduction rule i le obviou and more involved. Recall that a ubet M of vertice form a module if every vertex in M ha the ame neighbor outide M [Gallai 1967]. A module M of G i nontrivial if 1 < M < V (G). We oberve (ee Section 4.2) that a minimal forbidden et X of at leat 5 vertice i either fully contained in a module M or contain at mot one vertex of M. Moreover, if X M = {x}, then replacing x by any other vertex x M \ {x} in X reult in another minimal forbidden et. Thi permit u to branch on module, a decribed in the following reduction rule. Reduction 2. [Main] Let I = (G, k) be an intance where the graph G i prereduced, and a nontrivial module M that doe not induce a clique. (1 ) If every minimal forbidden et i contained in M, then return the intance (G[M], k). (2 ) If no minimal forbidden et i contained in M, then return the intance (G M, k), where G M i obtained from G by inerting edge to make G[M] a clique. (3 ) Otherwie, we olve three intance: I 1 = (G M, k M ), I 2 = (G[M], k 1), and I 3 = (G, k 1), where G i obtained from G by adding a clique M of (k + 1) vertice, connecting every pair of vertice u M and v N(M), and deleting M; letting Q 1, Q 2, and Q 3 be the olution of thee intance repectively, we return either Q 1 M or Q 2 Q 3 ( NO when Q 2 Q 3 > k), whichever i maller. That i, in the third cae we branch into two direction: the olution i obtained either a the union of M and the olution of I 1, or a the union of olution of I 2 and I 3. The two branche correpond to the two cae where the olution fully contain M or only a minimum interval deletion et to G[M] (i.e., Q 2 ), repectively. Note that in the econd branch, it can be hown that Q 3 i dijoint from M ; hence Q 2 Q 3 i indeed a ubet of V (G). Moreover, we have to clarify what the behavior of the reduction i if one or more of Q 1, Q 2, and Q 3 are NO. If Q 2 or Q 3 i NO, then we define Q 2 Q 3 to be NO a well. If one of Q 1 and Q 2 Q 3 i NO, we return the other one; if both of them are NO, we return NO a well. A graph on which neither reduction rule applie i called reduced; in uch a graph, every nontrivial module induce a clique. In Section 4, we prove the correctne of the reduction rule and that it can be checked in polynomial time if a reduction rule i applicable. Hence after exhautive application of the reduction, we may aume that the graph i reduced. The reduction are followed by a comprehenive tudy on reduced graph that yield two crucial combinatorial tatement. The firt tatement i on an AT {x, y, z} that are witneed by a minimal forbidden induced ubgraph W different from a hole. We ay that x i the hallow terminal if W N[x] i an induced path. We prove the hallow terminal x i implicial, i.e., N(x) induce a clique. Theorem 2.1. [Shallow terminal] All hallow terminal in a reduced graph are implicial. We ay that two hole are congenial to each other if each vertex of one hole i a neighbor of the other hole. It turn out that the hole are pairwie congenial in a reduced graph. Theorem 2.2. [Congenial hole] All hole in a reduced graph are congenial to each other.

6 0:6 Y. Cao and D. Marx We point out that circular-arc graph form an important example of graph where the hole are pairwie congenial. Indeed, all hole of a reduced graph induce a circular-arc graph, but uch a proof will not be given in thi paper, a it i unneceary for our purpoe here. One may refer to [van t Hof and Villanger 2013] for more intuition. PHASE 2: Breaking hole. A conequence of Theorem 2.2 i that if a vertex v i in a hole, then N[v] interect every hole and thu make a hole cover. Intuitively, thi ugget that a minimal hole cover ha to be very local in a certain ene. Indeed, by relating minimal hole cover in the reduced graph to minimal eparator in the ubgraph G N[v], we are able to etablih a quadratic bound on the number of minimal hole cover, and more importantly, a cubic time algorithm to contruct them. Theorem 2.3. [Hole cover] Every reduced graph of n vertice contain at mot n 2 minimal hole cover, and they can be enumerated in O(n 3 ) time. Any interval deletion et mut be a hole cover, and thu contain a minimal hole cover. Thi allow u to branch into at mot n 2 intance, in each of which the input graph i chordal. Note that thi branching tep i applied only once; hence only a polynomial factor will be induced in the running time. PHASE 3: Breaking AT. A all the hole have been broken, the graph i already chordal at the onet of the third phae. It hould be noted that, however, the graph might not be reduced, a new nontrivial non-clique module can be introduced with the deletion of a hole cover in Phae 2. In principle, we could rerun the reduction of Phae 1 to obtain a reduced intance, but there i no need to do o at thi point. The propertie that we need in thi phae are that graph i prereduced, chordal, and every hallow terminal i implicial (Theorem 2.1). We give a name to uch graph and compare it with previouly defined notion here. A graph i prereduced if Reduction 1 doe not apply. A prereduced graph i reduced if Reduction 2 doe not apply. A prereduced graph i nice if it i chordal and every hallow terminal in it i implicial. While both reduced graph and nice graph are prereduced, they are incomparable to each other. A only vertex deletion are applied after Phae 1, in the remainder of thi algorithm the graph i an induced ubgraph of that in a previou tep. In other word, once a hereditary property i obtained after Phae 1, it remain true thereafter. It i eay to verify that the three defining propertie of nice graph are all hereditary. On the one hand, after the end of Phae 1, a reduced graph i prereduced by definition, and according to Theorem 2.1, every hallow terminal in it i implicial. On the other hand, Phae 2 detroy all hole and the chordal property i obtained. Therefore, the graph become nice after Phae 2 and will remain nice till the end of our algorithm. The removal of all implicial vertice from a nice graph break all AT (Theorem 2.1), thereby yielding an interval graph. Thi implie that a nice graph ha a very pecial tructure: It ha a clique tree decompoition where the tree i a caterpillar, i.e., a path with degree-1 vertice attached to it. In other word, all vertice other than the hallow terminal can be arranged in a linear way, which greatly implifie the examination of interaction between AT. A a conequence, we can elect an AT that i minimal in a certain ene, and ingle out 10 vertice uch that there mut exit a minimum interval deletion et detroying thi AT with one of thee 10 vertice. We can therefore afely branch on removing one of thee 10 vertice. Theorem 2.4. [Nice graph] There i a 10 k n O(1) -time algorithm for interval deletion on nice graph.

7 Interval Deletion i Fixed-Parameter Tractable 0:7 Algorithm Interval-Deletion(G, k) input: a non-interval graph G and a poitive integer k. output: a minimum interval deletion et Q V (G) of ize k or NO. 1 Reduction 1: Let U be a minimal forbidden et of at mot 10 vertice; branch on deleting one vertex of U; \ the graph will then be prereduced and remain o hereafter; 2 Reduction 2: Let M be a nontrivial module of G not inducing a clique; 2.1 if all minimal forbidden et of G are contained in M then return Interval-Deletion(G[M], k); 2.2 ele if no minimal forbidden et i contained in M then return Interval-Deletion(G M, k), where edge are inerted to make G[M] a clique; 2.3 ele branch into three intance I 1, I 2, I 3 ; \ now the graph i reduced; 3 ue the algorithm of Theorem 2.3 to enumerate the at mot n 2 minimal hole cover of G; \ the graph will then be nice and remain o hereafter; 4 for each minimal hole cover HC do ue the algorithm of Theorem 2.4 to olve (G HC, k HC ); 5 return the mallet olution obtained, or NO if all olution are NO. Fig. 1: Outline of algorithm for interval deletion Putting together thee tep, the fixed-parameter tractability of interval deletion follow (ee Figure 1). Theorem 1.1 (retated). There i a 10 k n O(1) time algorithm for deciding whether or not there i a et of at mot k vertice whoe deletion make an n-vertex graph G an interval graph. Proof. The algorithm decribed in Figure 1 olve the problem by making recurive call to itelf, or calling the algorithm of Theorem 2.4 O(n 2 ) time. In the former cae, at mot 10 recurive call are made, all with parameter value at mot k 1. In the latter cae, the running time i 10 k n O(1). It follow that the total running time of the algorithm i 10 k n O(1). The paper i organized a follow. Section 3 et the definition and recall ome baic fact. Section 4 preent the detail of the firt phae. The next four ection are devoted to the proof of Theorem Section 5 and 6 put hallow terminal and congenial hole under thorough examination, and prove Theorem 2.1 and 2.2, repectively. Section 7 fully characterize minimal hole cover in reduced graph and prove Theorem 2.3. Section 8 preent the algorithm that detroy AT in nice graph and prove Theorem 2.4. Section 9 cloe thi paper by ome poible improvement and new direction. 3. PRELIMINARIES All graph dicued in thi paper hall alway be undirected and imple. A graph G i given by it vertex et V (G) and edge et E(G). If a pair of vertice v 1 and v 2 i connected by an edge, they are adjacent to each other, and denoted by v 1 v 2, otherwie nonadjacent and denoted by v 1 v 2. By v X we mean v i adjacent to at leat one vertex of the et X. Two vertex et X and Y are completely connected if x y for each pair of x X and y Y. A graph i complete if each pair of vertice are adjacent. A clique in a graph i a ubgraph that i complete, and a clique i maximal if it i not contained in another clique. A vertex i implicial if it neighbor induce a clique. A neighbor of a vertex i another vertex that i adjacent to it, and the et of neighborhood of a vertex v i denoted by N(v). The cloed neighborhood of v i defined a N[v] = N(v) {v}. Thi i generalized to a vertex et U, whoe cloed neighborhood and neighborhood are defined to be N[U] = v U N[v] and N(U) = N[U]\U. The notation N U (v) (N U [v]) tand for the neighbor of v in the et

8 0:8 Y. Cao and D. Marx U, i.e., N U (v) = N(v) U (N U [v] = N[v] U), regardle of whether v U or not. The ubgraph of a graph G induced by a ubet of vertice U i denoted by G[U], and G U i ued a a horthand for the ubgraph induced by V (G) \ U. A equence of ditinct vertice (v 0 v 1... v l ) uch that v i v i+1 for each 0 i < l i called a v 0 -v l path, whoe length i defined to be l. Vertice v 0 and v l are the end of the path, while other, {v 1,..., v l 1 }, are called inner vertice. If the end are ditinct and adjacent, i.e., l > 1 and v 0 v l, then (v 0 v 1... v l v 0 ) i called a cycle, whoe length i defined to be l+1. A an abue of notation, by u P (rep. u C) we mean that the vertex u appear in the path P (rep. cycle C), i.e., we ue P or C a the et of vertice in the path (rep. cycle). A chord in a path or cycle i an edge between two non-conecutive vertice in the path or cycle. It i worth noting that the edge v 0 v l, if exit, i a chord in the path (v 0 v 1... v l ), but not in the cycle (v 0 v 1... v l v 0 ). It i eay to verify that no hortet path can contain a chord, o between each pair of vertice of a connected graph there i a chordle path. A chordle cycle of length l, where l 4, i called an (l-)hole. A graph i chordal if it contain no hole, in other word, any cycle of length at leat 4 contain a chord. Chordal graph admit everal important and related characterization. A et S of vertice eparate x and y, and i called an x-y eparator if there i no x-y path in the ubgraph G S, and minimal x-y eparator if no proper ubet of S eparate x and y. For any pair of vertice x and y, a minimal x-y eparator i alo called a minimal eparator. A graph i chordal if and only if each minimal eparator in it induce a clique [Dirac 1961]. A nontrivial chordal graph contain at leat two implicial vertice, and there i at leat one implicial vertex in each component after the removal of any eparator. A tree T whoe node are the maximal clique of a graph G i a (maximal) clique tree of G if it atifie the following condition: any pair of adjacent node K i and K j define a minimal eparator that i K i K j ; for any vertex x V, the maximal clique containing x correpond to a ubtree of T. A graph i chordal if and only if it ha uch a clique tree. A clique tree of a graph G will be denoted by T (G), or T when the graph G i clear from the context. Without ditinguihing the node in a clique tree and the maximal clique in the graph G correponding to it, we ue K to denote both. A et of vertice i a minimal eparator of G if and only if it i the interection of K i and K j for ome edge K i K j in T [Buneman 1974]. Thi eparator, K i K j, i a minimal x-y eparator for any pair of vertice x K i \ K j and y K j \ K i. A interval graph are chordal, all aforementioned propertie alo apply to interval graph. Moreover, by the following characterization of Fulkeron and Gro, each interval graph ha a clique tree that i a path. Theorem 3.1 ([Fulkeron and Gro 1965]). A graph G i an interval graph if and only if the maximal clique of G can be linearly ordered uch that, for each vertex v, the maximal clique containing v occur conecutively. For a comprehenive treatment and for reference to the extenive literature on chordal graph and interval graph, one may refer to the monograph of Golumbic [2004] and the urvey of Brandtädt et al. [1999]. 4. REDUCTION RULES AND BRANCHING Thi ection dicue the reduction rule decribed in Section 2 in more detail Forbidden induced ubgraph Three vertice form an ateroidal triple, AT for hort, if each pair of them i connected by a path that avoid the neighborhood of the third one. We ue ateroidal witne (AW) to refer to a minimal induced ubgraph that i not a hole and contain an AT but none of it proper induced ubgraph doe. It hould be eay to check that an AW contain preciely one AT, and it vertice are the union of thee three defining path for thi triple; the three

9 Interval Deletion i Fixed-Parameter Tractable 0:9 t 1 t 1 t 1 t 1 t 2 c t 3 t 2 c t 3 t 2 t 3 t 2 t 3 (a) long claw (b) whipping top (c) net (d) tent c c 1 c 2 l r l r b 0 b 1 b 2 b i b d 1 b d (e) -AW ( : c : l, B, r) (d = B 3) b d+1 b 0 b 1 b 2 b i b d 1 b d b d+1 (f) -AW ( : c 1, c 2 : l, B, r) (d = B 2) Fig. 2: Minimal chordal ateroidal witnee (terminal are marked a quare). defining vertice will be called terminal of thi AW. It can be oberved from Figure 2 that the three terminal are the only implicial vertice of thi AW and they are nonadjacent to each other. Lekkerkerker and Boland [1962] oberved that a graph i an interval graph if and only if it i chordal and contain no AW. Not topping here, they rolled up their leeve and got their hand dirty by checking each poible forbidden induced ubgraph. Their effort brought the following le beautiful but more ueful characterization, here a minimal non-interval graph refer to a graph whoe every proper induced ubgraph i an interval graph but itelf i not. Theorem 4.1 ([Lekkerkerker and Boland 1962]). A minimal non-interval graph i either a hole or an AW depicted in Figure 2. Some remark are in order. Firt, it i eay to verify that a hole of ix or more vertice witnee an AT (pecifically, any three nonadjacent vertice from it) and i minimal, but following convention, we only refer to it a a hole, while reerve the term AW for graph lited in Figure 2. Second, the et of AW depicted in Figure 2 i not a literal copy of the original lit in [Lekkerkerker and Boland 1962], which contain neither net nor tent; they are viewed a -AW with d = 2 and -AW with d = 1, repectively. We ingle them out for the convenience of later preentation. To avoid ambiguitie, in thi paper we explicitly require a -AW (rep., -AW) to contain at leat 7 (rep., 8) vertice. Third, each of the four ubgraph in the firt row of Figure 2 conit of a contant number, 6 or 7, of vertice, and thu can be eaily located and dipoed of by tandard enumeration. For the purpoe of the current paper, we are mainly concerned with the two kind of AW in the econd row, whoe ize are unbounded. A - or -AW W contain a unique terminal, called the hallow terminal, uch that W N[] i an induced path. The neighbor() of the hallow terminal are the center(). The other two terminal are called bae terminal, and other vertice are called bae vertice. The whole et of bae vertice i called the bae. We ue ( : c : l, B, r) (rep., ( : c 1, c 2 : l, B, r)) to denote the -AW (rep., -AW) with hallow terminal, center c (rep., center c 1 and c 2 ), bae terminal l and r, and bae B = {b 1,..., b d }. For the ake of notational convenience, we will alo ue b 0 and b d+1 to refer to the bae terminal l and r, repectively, even though they are not part of the bae B. The center() and bae vertice are called non-terminal vertice. Clearly, Reduction 1 can be applied in polynomial time: we can find a minimal forbidden et of ize at mot 10 in polynomial time, e.g., by complete enumeration. There are way to improve thi, but optimizing the exponent i not the focu of thi paper. After the exhautive

10 0:10 Y. Cao and D. Marx application of Reduction 1, the graph i prereduced. By definition, any AW in a prereduced graph contain at leat 11 vertice, which rule out long claw, whipping top, net, and tent. Furthermore, the bae of a -AW (rep., -AW) in a prereduced graph contain at leat 7 (rep., 6) vertice. The purpoe of the following propoition and a detailed proof i twofold. Thee pecial tructure arie frequently in thi paper, and we do not want to repeat the ame argument again and again. The proof i exemplary in the ene that, by and large, mot proof of thi paper exploit a imilar contradictory argument: They explicitly contruct a forbidden induced ubgraph, either a mall AW or a hort hole, auming the property under dicuion doe not hold; becaue all graph dicued henceforth are prereduced, uch a contradiction will uffice to prove the deired property. Propoition 4.2. Let P = (v 0... v p ) be a chordle path of length p in a prereduced graph, and u be adjacent to every inner vertex of P. (1) If p 4 and u i alo adjacent to v 0 and v p, then N[v l ] N[u] for every 2 l p 2. (2) If p 3 and u i alo adjacent to v 0 and v p, then N[v l ] N[v l+1 ] N[u] for every 1 l p 2. (3) If p 4, then N[v l ] \ (N(v 1 ) N(v p 1 )) N[u] for every 2 l p 2. Proof. Suppoe to the contrary of tatement (1), there i a vertex x N[v l ]\N[u], then we how the exitence of a hort hole or mall AW in G, thu contradicting the aumption that G i prereduced. Note that x v i for any i l 2 or i l + 2, a otherwie, there i a 4-hole (uv i xv l u) (here v i v l becaue P i chordle). There i a 4-hole (uv l 1 xv l+1 u) when x i alo adjacent to both v l 1 and v l+1 ; a tent {u, v l 1, v l, x, v l+1, v l+2 } when x i adjacent to v l+1 but not v l 1 ; a tent {u, v l 2, v l 1, x, v l, v l+1 } when x i adjacent to v l 1 but not v l+1 ; or a whipping top {x, u, v l 2, v l 1, v l, v l+1, v l+2 } otherwie (x i only adjacent to v l in the path). Suppoe, for contradiction to tatement (2), x N[v l ] N[v l+1 ] \ N[u]. If x i adjacent to v l 1 or v l+2, then there i a 4-hole; otherwie, there i a tent {u, v l 1, v l, x, v l+1, v l+2 }. Statement (3) will follow from tatement (1) if u i alo adjacent to v 0 and v p ; hence we aume otherwie, and without lo of generality, u v 0. Suppoe to the contrary of tatement (3), there i a vertex x N[v l ] \ (N(v 1 ) N(v p 1 ) N[u]). If v 2 i the only inner vertex of P that i adjacent to x, then there i a 4-hole (xv 0 v 1 v 2 x) when x v 0 ; a 4-hole (xv 4 v 3 v 2 x) when x v 4 ; a net {v 0, v 1, x, v 2, u, v 4 } when x v 0, v 4 and u v 4 ; or a -AW (x : v 2 : v 0, v 1 uv 3, v 4 ) when x v 0, v 4 and u v 4. A ymmetric argument prove the cae when u v p and v p 2 i the only inner vertex of P that i adjacent to x. Other cae follow from tatement (1) and (2). Let X be a nonempty et of vertice. A vertex v i a common neighbor of X if it i adjacent to every vertex x X. We denote by N(X) the et of all common neighbor of X. It i eay to verify that in a prereduced graph, at leat one of X and N(X) induce a clique, a otherwie two nonadjacent vertice in N(X), together with two nonadjacent vertice in X, will induce a 4-hole. In particular, we have the following propoition. Propoition 4.3. Let X be a et of vertice of a prereduced graph that induce either a hole, an AW, or a path of length at leat 2. Then N(X) induce a clique Modular decompoition A ubet M of vertice form a module of G if all vertice in M have the ame neighborhood outide M. In other word, for any pair of vertice u, v M and vertex x M, u x if and only if v x. The et V (G) and all ingleton vertex et are module, called trivial.

11 Interval Deletion i Fixed-Parameter Tractable 0:11 A brief inpection how that no graph in Figure 2 ha any nontrivial module and thi i true alo for hole of length greater than 4: Propoition 4.4. Let M be a module, and X be a minimal forbidden et. If X > 4, then either X M, or M X 1. Indeed, the only minimal forbidden induced ubgraph of no more than 4 vertice i a 4- hole, of which the pair of nonadjacent vertice might belong to a module. Thi obervation allow u to prove the following tatement, which i the main combinatorial reaon behind the correctne of the branching in Reduction 2. Theorem 4.5. Let G be a graph that contain no 4-hole and M be a module of G. A minimum interval deletion et to G contain either all vertice of M, or only a minimum interval deletion et to G[M]. Proof. Let Q be a minimum interval deletion et to G uch that M Q; otherwie we are already done. To how that Q M = Q M i preciely a minimum interval deletion et to G[M], it uffice to how that for any minimum interval deletion et Q M to G[M], the et Q = (Q\Q M ) Q M i an interval deletion et to G: Trivially Q M i an interval deletion et to G[M]; if it i not minimum, then Q M > Q M, and Q > Q, which contradict the fact that Q i minimum. Suppoe the contrary and X i a minimal forbidden et in G Q. By contruction, Q M interect every minimal forbidden et in G[M], while Q \ Q M interect every minimal forbidden et in G M. Thu X interect both M and V (G) \ M. On the other hand, X > 4 a the graph i 4-hole free. According to Propoition 4.4, X M contain exactly one vertex; let it be x. Let x be a vertex in M \ Q, which i nonempty by the aumption M Q, and let X = X \ {x} {x }; it i immaterial whether x = x or not. The et X i dijoint from Q, and by definition of module, G[X ] and G[X] are iomorphic. In other word, X i a minimal forbidden et in G Q, which i impoible. Therefore, Q i a interval deletion et to G and thi finihe thi proof. To apply Reduction 2, we have to firt find a nontrivial module that i not a clique. For thi purpoe, we do not need to compute a modular decompoition tree of the graph. The imple algorithm decribed in Figure 3 i ufficient. Lemma 4.6. We can find in polynomial time a nontrivial module M uch that G[M] i not a clique, or report no uch a module exit. Proof. The algorithm decribed in Figure 3 find uch a module in a greedy manner. It tart from a pair of nonadjacent vertice u and v, and generate the module by adding vertice. Note that each vertex in the et X defined at tep 2.1 i a witne for the fact that M i not a module, in other word, M i a module only if X =. When a nonempty vertex et M i returned at tep 2.2, from the algorithm we can derive that X = and M V (G); hence M mut be a nontrivial module. Now it remain to how that a long a there i a nontrivial non-clique module U in the graph, the algorithm i guaranteed to return a nonempty et (not necearily U itelf). A U doe not induce a clique, it contain a pair of nonadjacent vertice u and v, which hall be conidered in ome iteration of the for-loop. In thi iteration, initially M U, and by induction we are able to how that no vertex of V (G) \ U can be included in X during thi iteration; hence M U will remain an invariant. A a conequence, a ubet M that atifie {u, v} M U i returned. Indeed, one can eaily verify that the module found a above i the incluive-wie minimal one containing both u and v. We are now ready to explain the application of Reduction 2 and prove it correctne.

12 0:12 Y. Cao and D. Marx for each pair of nonadjacent vertice u and v do 1 M = {u, v}; 2 while M V (G) do 2.1 X = {x M : 0 < N M (x) < M }; 2.2 if X = then return M; 2.3 ele M = M X; return. \ there i no uch a module Fig. 3: Algorithm Find-Module Lemma 4.7. Reduction 2 i correct and it can be checked in polynomial time whether Reduction 2 (and which cae of it) i applicable. Proof. The correctne of the reduction i clear in cae 1: removing the vertice of V (G) \ M doe not make the problem any eaier, a thee vertice do not participate in any minimal forbidden et. In cae 2, the correctne of the reduction follow from the fact that G and G M have the ame et of minimal forbidden et. Note that a clique i an interval graph, and more importantly, the inertion of edge to make M a clique neither break the modularity of M nor introduce any new 4-hole; thu Propoition 4.4 i applicable to G M. A M induce an interval graph in both G and G M, if X i a minimal forbidden et of G or G M, then Propoition 4.4 implie that X contain at mot one vertex of M. In other word, the inertion of edge ha no effect on any minimal forbidden et, which mean that Q i an interval deletion et to G if and only if it i an interval deletion et to G M. The correctne of cae 3 can be argued uing Theorem 4.5, which tate the two poibilitie of any interval deletion et to G with repect to M. In particular, the two branche of cae 3 correpond to thee two cae. The firt branch i traightforward: we imply remove all vertice of M from the graph and olve the intance I 1 = (G M, k M ). It i the econd branch (where we aume M Q) that need more explanation. Recall that by contruction of I 3, the et M i a module of G and induce an interval graph. It i clear that either olution Q 2 or Q 3 being NO will rule out the exitence of an interval deletion et of G that doe not fully contain M. Hence we may aume Q 2 and Q 3 are minimum interval deletion et of I 2 and I 3, repectively; and Q = Q 2 Q 3. Note that both Q 2 and Q 3 are upper bounded by k 1. Claim 1. Set Q i an interval deletion et of G. Proof. According to Theorem 4.5, if Q 3 interect M, which i a module of G, then it mut contain all (k + 1) vertice in M, 1 i.e., Q 3 > k; a contradiction. Therefore, Q 3 M =, which mean Q V (G). Suppoe that there i a minimal forbidden et X of G dijoint from Q. It cannot be fully contained in M, a Q 2 Q i an interval deletion et of G[M]. Then by Propoition 4.4, X contain exactly one vertex x of M and X = X \ {x} {x } i alo a minimal forbidden et of G for any x M. Since Q 3 i an interval deletion et of G dijoint from M, it ha to contain a vertex of X \{x } = X \{x}; a contradiction. Claim 2. Q. Set Q i not larger than the mallet interval deletion et Q atifying M Proof. Suppoe that Q i an interval deletion et of G of ize at mot k with M Q ; let Q 2 = Q M and Q 3 = Q \ M. We claim that Q 2 and Q 3 are interval deletion et of I 2 and I 3, repectively. Firt, we argue that Q 2 and Q 3 are not empty; hence both of them 1 Indeed, min(k + 1, N(M) ) vertice will uffice for our bookkeeping purpoe, and an alternative way to thi i to add only one vertex but mark it a forbidden.

13 Interval Deletion i Fixed-Parameter Tractable 0:13 have ize at mot k 1. The aumption that G[M] i not an interval graph implie Q 2. By aumption, M Q, thu there i a vertex x M \ Q. Now Q 3 = would imply that G (M \{x}) i an interval graph, that i, there i no minimal forbidden et containing only one vertex of M, and it follow that we hould have been in Cae 1. Since Q 2 k 1, it i clear that Q 2 i a olution of intance I 2 = (G[M], k 1). The only way Q 3 i not a olution of I 3 i that there i a minimal forbidden et X containing a vertex of the (k + 1)-clique introduced to replace M. A thi (k + 1)-clique i a module, Propoition 4.4 implie that X contain exactly one vertex y of thi clique. But in thi cae X = X \ {y} {x} (where x i a vertex of M \ Q ) i a minimal forbidden et dijoint from Q, a contradiction. Thu Q Q follow from the fact that both Q 2 and Q 3 are minimum. A a conequence of Claim 2, if Q > k, then there cannot be an interval deletion et of ize no more than k that doe not fully include M. Thi finihe the proof of the correctne of Reduction 2. On the applicability of Reduction 2, we firt ue Lemma 4.6 to find a nontrivial module that doe not induce a clique. If uch a module M i found, then Reduction 2 i applicable, and it remain to figure out which cae hould apply by checking the condition in order. To check whether cae 1 hold, we need to check if there i a minimal forbidden et X not contained in M. By Propoition 4.4, uch an X, if exit, contain at mot one vertex x from M; and x can be replaced by any other vertex of M. Therefore, it uffice to pick any vertex x M, and tet in linear time whether G (M \ {x}) i an interval graph. If it i not an interval graph, then there i a minimal forbidden et X not contained in M (a it contain at mot one vertex of M). Otherwie, G (M \ {x}) i an interval graph for every x M, and there i no uch X; hence cae 1 hold. To check whether cae 2 hold, oberve that the condition there i no minimal forbidden et contained in M i equivalent to aying that G[M] i an interval graph, which can be checked in linear time. In all remaining cae, we are in cae SHALLOW TERMINALS Thi ection prove Theorem 2.1 by howing that each hallow terminal i contained in a module whoe neighborhood induce a clique. Thi module either i trivial (coniting of only thi hallow terminal), or induce a clique (after the application of Reduction 2). Therefore, thi hallow terminal i alway implicial. Recall that an AW in a prereduced graph G ha to be a - or -AW. Let u tart from a thorough crutiny of neighbor of it hallow terminal, which, by definition, i dijoint from the bae and bae terminal. Lemma 5.1. Let W be an AW in a prereduced graph. Every common neighbor x of the bae B i adjacent to the hallow terminal. Proof. The center() of W are alo common neighbor of B, and hence according to Propoition 4.3, they are adjacent to x. Suppoe, for contradiction, x N(B) \ N(). If W i a -AW, then there i (ee the firt row of Figure 4) a whipping top {, c, l, b 1, x, b d, r} centered at c when x l, r; a net {, c, l, b 1, r, x} when x r but x l (imilarly for x l but x r); or a -AW ( : c : l, b 1 xb d, r) when x l, r. If W i a -AW, then there i (ee the econd row of Figure 4) a tent {x, c 1, b 1,, b d, c 2 } when x l, r; a -AW ( : c 1, c 2 : l, b 1 x, r) when x r but x l (imilarly for x l but x r); or a -AW ( : c 1, c 2 : l, b 1 xb d, r) when x l, r. A none of thee tructure can exit in a prereduced graph, thi lemma i proved. Lemma 5.2. terminal. Let W be an AW in a prereduced graph, and x be adjacent to the hallow (1) Then x i alo adjacent to the center() of W (different from x).

14 0:14 Y. Cao and D. Marx c c c l b 1 b d r x (a) -AW, x l, r c 1 c 2 l b 1 b d r x (b) -AW, x l, and x r c 1 c 2 l b 1 b d r x (c) -AW, x l, r c 1 c 2 l b 1 b d r l b 1 b d r l b 1 b d r x x x (d) -AW, x l, r (e) -AW, x l, and x r (f) -AW, x l, r Fig. 4: Adjacency between a common neighbor x of B and [Lemma 5.1]. (2) Claifying x with repect to it adjacency to the bae B, we have the following categorie: (full) x i adjacent to every bae vertex. Then x i alo adjacent to every vertex in N() \ {x}. (partial) x i adjacent to ome, but not all bae vertice. Then there i an AW whoe hallow terminal i, one center i x, and bae i a proper ub-path of B. (none) x i adjacent to no bae vertex. Then x i adjacent to neither bae terminal, and thu replacing the hallow terminal of W by x make another AW. Proof. Suppoe to the contrary of tatement (1), x c if W i a -AW or (without lo of generality) x c 2 if W i a -AW. If x b i for ome 1 i d then there i a 4-hole (xcb i x) or (xc 2 b i x) (See Figure 5(a)). Hence we may aume x B. (See Figure 5(b,c,d,e).) There i a 5-hole (xcb 1 lx) or (xcb d rx) if W i a -AW, and x l or x r, repectively; a 5-hole (xc 2 b 1 lx) or 4-hole (xc 2 rx) if W i a -AW, and x l or x r, repectively; a long claw {x,, c, b 1, l, b d, r} if W i a -AW and x l, r; a net {x,, l, c 1, r, c 2 } if W i a -AW and x c 1, l, r; or a whipping top {r, c 2,, x, c 1, l, b 1 } centered at c 2 if W i a -AW and x l, r, but x c 1. Neither of thee cae i poible, and thu tatement (1) i proved. For tatement (2), let u handle category none firt. Note that x, nonadjacent to B, cannot be a center of W. If x l, then there i a 4-hole (xcb 1 lx) or (xc 2 b 1 lx) when W i a -AW or -AW, repectively. A ymmetrical argument will rule out x r. Now that x i adjacent to the center() but neither bae terminal nor bae vertice of W, then (x : c : l, B, r) (rep., (x : c 1, c 2 : l, B, r)) make another -AW (rep., -AW). Aume now that x i in category full. Suppoe for contradiction that x v for ome v N() \ {x}. We have already proved in tatement (1) that v and x are adjacent to the center() of W (different from them). In particular, if one of v and x i a center, then they

15 Interval Deletion i Fixed-Parameter Tractable 0:15 x x c 1 c 2 c 1 c 2 (a) x B (b) x B but x {l, r} x x x c c 1 c 2 c 1 c 2 (c) x but x B (d) x but x B (e) x c 1 but x c 2, l Fig. 5: Adjacency between a neighbor x of and center [Lemma 5.2]. x c x c x c l b d r l b d r l b d r b 1 b 1 b 3 b 1 b 3 (a) N B (x) = {b 1 } and x l. (b) N B (x) = {b 1 } and x l. (c) N B (x) = {b 1, b 2 }. x c x c l b 1 b d r l b 1 b d r b i 2 b i b i+2 (d) N B (x) = {b i } (1 < i < d). b i 1 b j+1 (e) N B (x) = {b i, b i+1 } (1 < i < d 1). Fig. 6: Vertex x in category partial w.r.t. W [Lemma 5.2]. are adjacent. Therefore, we can aume that v and x are not center. If v b i for ome 1 i d, then there i a 4-hole (xvb i x). Otherwie, v B, and it i in category none. Let W be the AW obtained by replacing in W by v; then x v follow from Lemma 5.1. Finally, aume that x i in category partial, that i, x B, but x b i for ome 1 i d. In thi cae, we contruct the claimed AW a follow. A the cae x l but x r i ymmetric to x l but x r, it i ignored in the following, i.e., we aume that x r only if x l. Let p be the mallet index uch that x b p, and q be the mallet index uch