Polynomial Kernel for Distance-Hereditary Deletion
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Finding Articulation Points and Bridges Articulation Points Articulation Point
Finding Articulation Points and Bridges Articulation Points Articulation Point Articulation Point A vertex v is an articulation point (also called cut vertex) if removing v increases the number of connected components. A graph with two articulation points. 3 / 1 Articulation Points Given I An undirected, connected graph G = (V; E) I A DFS-tree T with the root r Lemma A DFS on an undirected graph does not produce any cross edges. Conclusion I If a descendant u of a vertex v is adjacent to a vertex w, then w is a descendant or ancestor of v. 4 / 1 Removing a Vertex v Assume, we remove a vertex v 6= r from the graph. Case 1: v is an articulation point. I There is a descendant u of v which is no longer reachable from r. I Thus, there is no edge from the tree containing u to the tree containing r. Case 2: v is not an articulation point. I All descendants of v are still reachable from r. I Thus, for each descendant u, there is an edge connecting the tree containing u with the tree containing r. 5 / 1 Removing a Vertex v Problem I v might have multiple subtrees, some adjacent to ancestors of v, and some not adjacent. Observation I A subtree is not split further (we only remove v). Theorem A vertex v is articulation point if and only if v has a child u such that neither u nor any of u's descendants are adjacent to an ancestor of v. Question I How do we determine this efficiently for all vertices? 6 / 1 Detecting Descendant-Ancestor Adjacency Lowpoint The lowpoint low(v) of a vertex v is the lowest depth of a vertex which is adjacent to v or a descendant of v. -
The Strong Perfect Graph Theorem
Annals of Mathematics, 164 (2006), 51–229 The strong perfect graph theorem ∗ ∗ By Maria Chudnovsky, Neil Robertson, Paul Seymour, * ∗∗∗ and Robin Thomas Abstract A graph G is perfect if for every induced subgraph H, the chromatic number of H equals the size of the largest complete subgraph of H, and G is Berge if no induced subgraph of G is an odd cycle of length at least five or the complement of one. The “strong perfect graph conjecture” (Berge, 1961) asserts that a graph is perfect if and only if it is Berge. A stronger conjecture was made recently by Conforti, Cornu´ejols and Vuˇskovi´c — that every Berge graph either falls into one of a few basic classes, or admits one of a few kinds of separation (designed so that a minimum counterexample to Berge’s conjecture cannot have either of these properties). In this paper we prove both of these conjectures. 1. Introduction We begin with definitions of some of our terms which may be nonstandard. All graphs in this paper are finite and simple. The complement G of a graph G has the same vertex set as G, and distinct vertices u, v are adjacent in G just when they are not adjacent in G.Ahole of G is an induced subgraph of G which is a cycle of length at least 4. An antihole of G is an induced subgraph of G whose complement is a hole in G. A graph G is Berge if every hole and antihole of G has even length. A clique in G is a subset X of V (G) such that every two members of X are adjacent. -
Enumeration of Unlabeled Graph Classes a Study of Tree Decompositions and Related Approaches
Enumeration of unlabeled graph classes A study of tree decompositions and related approaches Jessica Shi Advisor: Jérémie Lumbroso Independent Work Report Fall, 2015 Abstract In this paper, we study the enumeration of certain classes of graphs that can be fully charac- terized by tree decompositions; these classes are particularly significant due to the algorithmic improvements derived from tree decompositions on classically NP-complete problems on these classes [12, 7, 17, 35]. Previously, Chauve et al. [6] and Iriza [26] constructed grammars from the split decomposition trees of distance hereditary graphs and 3-leaf power graphs. We extend upon these results to obtain an upper bound grammar for parity graphs. Also, Nakano et al. [25] used the vertex incremental characterization of distance hereditary graphs to obtain upper bounds. We constructively enumerate (6;2)-chordal bipartite graphs, (C5, bull, gem, co-gem)-free graphs, and parity graphs using their vertex incremental characterization and extend upon Nakano et al.’s results to analytically obtain upper bounds of O7n and O11n for (6;2)-chordal bipartite graphs and (C5, bull, gem, co-gem)-free graphs respectively. 1. Introduction 1.1. Context The technique of decomposing graphs into trees has been an object of significant interest due to its applications on classical problems in graph theory. Indeed, many graph theoretic problems are inherently difficult due to the lack of recursive structure in graphs, and recursion has historically offered efficient solutions to challenging problems. In this sense, classifying graphs in terms of trees is of particular interest, since it associates a recursive structure to these graphs. -
P 4-Colorings and P 4-Bipartite Graphs Chinh T
P_4-Colorings and P_4-Bipartite Graphs Chinh T. Hoàng, van Bang Le To cite this version: Chinh T. Hoàng, van Bang Le. P_4-Colorings and P_4-Bipartite Graphs. Discrete Mathematics and Theoretical Computer Science, DMTCS, 2001, 4 (2), pp.109-122. hal-00958951 HAL Id: hal-00958951 https://hal.inria.fr/hal-00958951 Submitted on 13 Mar 2014 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Discrete Mathematics and Theoretical Computer Science 4, 2001, 109–122 P4-Free Colorings and P4-Bipartite Graphs Ch´ınh T. Hoang` 1† and Van Bang Le2‡ 1Department of Physics and Computing, Wilfrid Laurier University, 75 University Ave. W., Waterloo, Ontario N2L 3C5, Canada 2Fachbereich Informatik, Universitat¨ Rostock, Albert-Einstein-Straße 21, D-18051 Rostock, Germany received May 19, 1999, revised November 25, 2000, accepted December 15, 2000. A vertex partition of a graph into disjoint subsets Vis is said to be a P4-free coloring if each color class Vi induces a subgraph without a chordless path on four vertices (denoted by P4). Examples of P4-free 2-colorable graphs (also called P4-bipartite graphs) include parity graphs and graphs with “few” P4s like P4-reducible and P4-sparse graphs. -
Induced Path Factors of Regular Graphs Arxiv:1809.04394V3 [Math.CO] 1
Induced path factors of regular graphs Saieed Akbari∗ Daniel Horsley† Ian M. Wanless† Abstract An induced path factor of a graph G is a set of induced paths in G with the property that every vertex of G is in exactly one of the paths. The induced path number ρ(G) of G is the minimum number of paths in an induced path factor of G. We show that if G is a connected cubic graph on n > 6 vertices, then ρ(G) 6 (n − 1)=3. Fix an integer k > 3. For each n, define Mn to be the maximum value of ρ(G) over all connected k-regular graphs G on n vertices. As n ! 1 with nk even, we show that ck = lim(Mn=n) exists. We prove that 5=18 6 c3 6 1=3 and 3=7 6 c4 6 1=2 and that 1 1 ck = 2 − O(k− ) for k ! 1. Keywords: Induced path, path factor, covering, regular graph, subcubic graph. Classifications: 05C70, 05C38. 1 Introduction We denote the path of order n by Pn. A subgraph H of a graph G is said to be induced if, for any two vertices x and y of H, x and y are adjacent in H if and only if they are adjacent in G. An induced path factor (IPF) of a graph G is a set of induced paths in G with the property that every vertex of G is in exactly one of the paths. We allow paths of any length in an IPF, including the trivial path P1. -
Polynomial-Time Algorithms for the Longest Induced Path and Induced Disjoint Paths Problems on Graphs of Bounded Mim-Width∗†
Polynomial-Time Algorithms for the Longest Induced Path and Induced Disjoint Paths Problems on Graphs of Bounded Mim-Width∗† Lars Jaffke‡1, O-joung Kwon§2, and Jan Arne Telle3 1 Department of Informatics, University of Bergen, Norway [email protected] 2 Logic and Semantics, Technische Universität Berlin, Berlin, Germany [email protected] 3 Department of Informatics, University of Bergen, Norway [email protected] Abstract We give the first polynomial-time algorithms on graphs of bounded maximum induced matching width (mim-width) for problems that are not locally checkable. In particular, we give nO(w)-time algorithms on graphs of mim-width at most w, when given a decomposition, for the following problems: Longest Induced Path, Induced Disjoint Paths and H-Induced Topological Minor for fixed H. Our results imply that the following graph classes have polynomial-time algorithms for these three problems: Interval and Bi-Interval graphs, Circular Arc, Per- mutation and Circular Permutation graphs, Convex graphs, k-Trapezoid, Circular k-Trapezoid, k-Polygon, Dilworth-k and Co-k-Degenerate graphs for fixed k. 1998 ACM Subject Classification F.2.2 Nonnumerical Algorithms and Problems, Computations on discrete structures, G.2.2 Graph Theory, Graph algorithms Keywords and phrases graph width parameters, dynamic programming, graph classes, induced paths, induced topological minors Digital Object Identifier 10.4230/LIPIcs.IPEC.2017.21 1 Introduction Ever since the definition of the tree-width of graphs emerged from the Graph Minors project of Robertson and Seymour, bounded-width structural graph decompositions have been a successful tool in designing fast algorithms for graph classes on which the corresponding width-measure is small. -
Algorithmic Graph Theory Part III Perfect Graphs and Their Subclasses
Algorithmic Graph Theory Part III Perfect Graphs and Their Subclasses Martin Milanicˇ [email protected] University of Primorska, Koper, Slovenia Dipartimento di Informatica Universita` degli Studi di Verona, March 2013 1/55 What we’ll do 1 THE BASICS. 2 PERFECT GRAPHS. 3 COGRAPHS. 4 CHORDAL GRAPHS. 5 SPLIT GRAPHS. 6 THRESHOLD GRAPHS. 7 INTERVAL GRAPHS. 2/55 THE BASICS. 2/55 Induced Subgraphs Recall: Definition Given two graphs G = (V , E) and G′ = (V ′, E ′), we say that G is an induced subgraph of G′ if V ⊆ V ′ and E = {uv ∈ E ′ : u, v ∈ V }. Equivalently: G can be obtained from G′ by deleting vertices. Notation: G < G′ 3/55 Hereditary Graph Properties Hereditary graph property (hereditary graph class) = a class of graphs closed under deletion of vertices = a class of graphs closed under taking induced subgraphs Formally: a set of graphs X such that G ∈ X and H < G ⇒ H ∈ X . 4/55 Hereditary Graph Properties Hereditary graph property (Hereditary graph class) = a class of graphs closed under deletion of vertices = a class of graphs closed under taking induced subgraphs Examples: forests complete graphs line graphs bipartite graphs planar graphs graphs of degree at most ∆ triangle-free graphs perfect graphs 5/55 Hereditary Graph Properties Why hereditary graph classes? Vertex deletions are very useful for developing algorithms for various graph optimization problems. Every hereditary graph property can be described in terms of forbidden induced subgraphs. 6/55 Hereditary Graph Properties H-free graph = a graph that does not contain H as an induced subgraph Free(H) = the class of H-free graphs Free(M) := H∈M Free(H) M-free graphT = a graph in Free(M) Proposition X hereditary ⇐⇒ X = Free(M) for some M M = {all (minimal) graphs not in X} The set M is the set of forbidden induced subgraphs for X. -
Long Induced Paths in Graphs∗ Arxiv:1602.06836V2 [Math.CO] 1
Long induced paths in graphs∗ Louis Esperety Laetitia Lemoinez Fr´ed´ericMaffrayx December 2, 2016 Abstract We prove that every 3-connected planar graph on n vertices contains an induced path on Ω(log n) vertices, which is best possible and improves the best known lower bound by a multiplicative factor of log log n. We deduce that any planar graph (or more generally, any graph embeddable on a fixed surface) with a path on n vertices, also contains an induced path on Ω(plog n) vertices. We conjecture that for any k, there is a positive constant c(k) such that any k-degenerate graph with a path on n vertices also contains an induced path on Ω((log n)c(k)) vertices. We provide examples showing that this order of magnitude would be best possible (already for chordal graphs), and prove the conjecture in the case of interval graphs. 1 Introduction A graph contains a long induced path (i.e., a long path as an induced subgraph) only if it contains a long path. However, this necessary condition is not sufficient, as shown by complete graphs and complete bipartite graphs. On the other hand, it was proved by Atminas, Lozin and Ragzon [2] that if a graph G contains a long path, but does not contain a large complete graph or complete bipartite graph, then G contains a long induced path. Their proof uses several applications of Ramsey theory, and the resulting bound on the size of a long induced path is thus quantitatively weak. The specific case of k-degenerate graphs (graphs such that any subgraph contains a arXiv:1602.06836v2 [math.CO] 1 Dec 2016 vertex of degree at most k) was considered by Neˇsetˇriland Ossona de Mendez in [6]. -
Classes of Perfect Graphs
This paper appeared in: Discrete Mathematics 306 (2006), 2529-2571 Classes of Perfect Graphs Stefan Hougardy Humboldt-Universit¨atzu Berlin Institut f¨urInformatik 10099 Berlin, Germany [email protected] February 28, 2003 revised October 2003, February 2005, and July 2007 Abstract. The Strong Perfect Graph Conjecture, suggested by Claude Berge in 1960, had a major impact on the development of graph theory over the last forty years. It has led to the definitions and study of many new classes of graphs for which the Strong Perfect Graph Conjecture has been verified. Powerful concepts and methods have been developed to prove the Strong Perfect Graph Conjecture for these special cases. In this paper we survey 120 of these classes, list their fundamental algorithmic properties and present all known relations between them. 1 Introduction A graph is called perfect if the chromatic number and the clique number have the same value for each of its induced subgraphs. The notion of perfect graphs was introduced by Berge [6] in 1960. He also conjectured that a graph is perfect if and only if it contains, as an induced subgraph, neither an odd cycle of length at least five nor its complement. This conjecture became known as the Strong Perfect Graph Conjecture and attempts to prove it contributed much to the developement of graph theory in the past forty years. The methods developed and the results proved have their uses also outside the area of perfect graphs. The theory of antiblocking polyhedra developed by Fulkerson [37], and the theory of modular decomposition (which has its origins in a paper of Gallai [39]) are two such examples. -
Partitioning a Graph Into Disjoint Cliques and a Triangle-Free
Partitioning a Graph into Disjoint Cliques and a Triangle-free Graph Faisal N. Abu-Khzam, Carl Feghali, Haiko M¨uller January 6, 2015 Abstract A graph G =(V,E) is partitionable if there exists a partition {A, B} of V such that A induces a disjoint union of cliques and B induces a triangle- free graph. In this paper we investigate the computational complexity of deciding whether a graph is partitionable. The problem is known to be NP-complete on arbitrary graphs. Here it is proved that if a graph G is bull-free, planar, perfect, K4-free or does not contain certain holes then deciding whether G is partitionable is NP-complete. This answers an open question posed by Thomass´e, Trotignon and Vuˇskovi´c. In contrast a finite list of forbidden induced subgraphs is given for partitionable cographs. 1 Introduction A graph G = (V, E) is monopolar if there exists a partition {A, B} of V such that A induces a disjoint union of cliques and B forms an independent set. The class of monopolar graphs has been extensively studied in recent years. It is known that deciding whether a graph is monopolar is NP-complete [18], even when restricted to triangle-free graphs [6] and planar graphs [20]. In contrast the problem is tractable on several graph classes: a non-exhaustive list includes cographs [17], polar permutation graphs [15], chordal graphs [16], line graphs [7] and several others [8]. A graph is (k,l)-partitionable if it can be partitioned arXiv:1403.5961v6 [cs.CC] 4 Jan 2015 in up to k cliques and l independent sets with k + l ≥ 1. -
Absolute Retracts of Split Graphs
DISCRETE MATHEMATICS ELSEIVIER Discrete Mathematics 134 (1994) 75-84 Absolute retracts of split graphs Sandi Klaviar ’ University of’ Maribor, PF, KoroSka cesta 160, 62000 Maribor, Slosenia Received 20 February 1992; revised 20 July 1992 Abstract It is proved that a split graph is an absolute retract of split graphs if and only if a partition of its vertex set into a stable set and a complete set is unique or it is a complete split graph. Three equivalent conditions for a split graph to be an absolute retract of the class of all graphs are given. It is finally shown that a reflexive split graph G is an absolute retract of reflexive split graphs if and only if G has no retract isomorphic to some J,, n B 3. Here J, is the reflexive graph with vertex set {?c~,x*, . , x,,y,, y,, , y,} in which the vertices x1, x2, , x, are mutually adjacent and thevertexy,isadjacent to x~,x~,...,x~_~,x~+~,...,x,. 1. Introduction The main motivation for this paper is the investigation [l] by Bandelt et al. in which the internal structure of the absolute retracts of bipartite graphs is clarified. They proved several nice characterizations of bipartite absolute retracts and applied these graphs to the competitive location theory. However, the investigation of the absolute retracts of bipartite graphs started as early as in 1972 with the Ph.D. Thesis of Hell [S], who gave, among others, two characterizations of the absolute retracts of bipartite graphs. Split graphs are ‘half-way’ between bipartite graphs and their complements. -
Solutions to Exercises 7
Discrete Mathematics Lent 2009 MA210 Solutions to Exercises 7 (1) The complete bipartite graph Km;n is defined by taking two disjoint sets, V1 of size m and V2 of size n, and putting an edge between u and v whenever u 2 V1 and v 2 V2. (a) How many edges does Km;n have? Solution. Every vertex of V1 is adjacent to every vertex of V2, hence the number of edges is mn. (b) What is the degree sequence of Km;n? Solution. Every vertex of V1 has degree n because it is adjacent to every vertex of V2. Similarly, every vertex of V2 has degree m because it is adjacent to every vertex of V2. So the degree sequence of Km;n consists of m n’s and n m’s listed in non-increasing order. If m ≥ n, then the degree sequence is (m; : : : ; m; n; : : : ; n): | {z } | {z } n m If m < n, then the degree sequence is (n; : : : ; n; m; : : : ; m): | {z } | {z } m n (c) Which complete bipartite graphs Km;n are connected? Solution. Take any m; n ≥ 1. For any vertex x 2 V1, y 2 V2, the pair xy is an edge, so x; y is a walk from x to y. For vertices x; y 2 V1, x 6= y, take any w 2 V2. The pairs xw; wy are edges, so x; w; y is a walk from x to y. For vertices x; y 2 V2, x 6= y, take any w 2 V1. The pairs xw; wy are edges, so x; w; y is a walk from x to y.