DOCSLIB.ORG

Chordal Graphs: Theory and Algorithms

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Chordal Graphs: Theory and Algorithms Chordal Graphs: Theory and Algorithms 1 Chordal graphs Chordal graph : Every cycle of four or more vertices has a chord in it, i.e. there is an edge between two non consecutive vertices of the cycle. Also called rigid circuit graphs, Perfect Elimination Graphs, Triangulated Graphs, monotone transitive graphs. 2 Subclasses of Chordal Graphs Trees K-trees ( 1-tree is tree) Kn: Complete Graph Block Graphs Bipartite Graph with bipartition X and Y ( not necessarily Chordal ) Split Graphs: Make one part of Bipartite graph complete Interval Graphs 3 Subclasses of Chordal Graphs … Rooted Directed Path Graphs Directed Path Graphs Path Graphs Strongly Chordal Graphs Doubly Chordal Graphs 4 Research Issues in Trees Computing Achromatic number in tree is NP-Hard Conjecture: Every tree is Graceful L(2,1)-labeling number of a tree with maximum degree is either +1 or +2. Characterize trees having L(2,1)-labeling number +1 5 Why Study a Special Graph Class? Reasons The Graph Class arises from applications The graph class posses some interesting structures that helps solving certain hard but important problems restricted to this class some interesting meaningful theory can be developed in this class. All of these are true for chordal Graphs. Hence study Chordal Graphs. See the book: Graph Classes (next slide) 6 7 Characterizing Property: Minimal separators are cliques ( Dirac 1961) Characterizing Property: Every minimal a-b vertex separator is a clique ( Complete Subgraph) Minimal a-b Separator: S V is a minimal vertex separator if a and b lie in different components of G-S and no proper subset of S has this property. 8 Lemma 1 A vertex x of G is called simplicial if its adjacency set Adj(x) is a clique (not necessarily maximal) Lemma 1[Dirac 1961]: Each chordal graph has a simplicial vertex and if G is not a clique it has two non adjacent simplicial vertices. 9 Proof of Lemma 1 If G is a clique, done. If not, assume that G has two non-adjacent nodes a and b and that the lemma is true to all graphs with fewer vertices than G. Let S be a minimal vertex separator for a and b. Let Ga and Gb be the connected components of a and b respectively. 1 0 Proof of Lemma 1 (Cont.) S is a clique – if not, then there exists x and y such that there is no edge between them, and because S is minimal vertex separator there is a cycle <a,…,x,…,b,…,y,…,a>. It is easy to show that there exist a minimal cycle with length no less then 4 with no chords in it, contradiction. 1 1 Proof of Lemma 1 (Cont.) S is a clique. GA+S is smaller than G therefore by induction the lemma holds, i.e. GA+S is a clique or has two non adjacent simplicial vertices, one of each must be in GA. Any simplicial vertex in GA is a simplicial vertex in G because all elements of Adj(A) are inside GA+S. Thus from GA and GB we get two simplicial vertices in G. 1 2 Characterizing Property: PEO ( Fulkerson and Gross 1965) Lemma: (Dirac 1961) Every chordal graph has a simplicial vertex. If G is not complete, then it has two non-adjacent simplicial vertices. Simplicial: G[NG(v)] is complete Theorem: G is chordal iff it has a Perfect Elimination Ordering 2 4 (1,2,3,4,5) is a PEO of G 5 1 3 G PEO: (v1,v2,…,vn) is a PEO if vi is a simplicial vertex of G[{vi,vi+1,…,vn}], 1 ≤ i ≤n. 13 PEO Illustration Perfect Elimination order : necessary and sufficient condition for Chordal graphs Simplicial vertex : along with their neighbors form a clique 14 Theorem 7 Definition: The following are equivalent: 1. G is chordal 2. The edges of G can be directed acyclically so that every pair of converging arrows comes from two adjacent vertices. 3. G has a perfect vertex elimination scheme. 4. There is a tree T with maximal cliques of G as vertices. Any two cliques containing v are either adjacent in T or connected by a path of cliques that contain v. 1 5 Proof of Theorem 7 (1→2) By lemma 1 every chordal graph has a simplicial vertex Direct all of its edges to it Repeat the process with the rest of the graph (can be done because being chordal is hereditary) From the definition of simplicial vertex corollary that each two converging edges coming from adjacent vertices. 1 6 Proof of Theorem 7 (2→1) Let there be a cycle of size larger than 3. There exist a valid direction which is acyclic There are two converging edges, which come from two adjacent vertices The cycle has a chord. 1 7 Perfect Vertex Elimination Let G=(V,E) be an undirected graph and let σ=[v1,v2,…,vn] be an ordering of the vertices. We say that σ is a perfect vertex elimination scheme if for each i: Xi = { vj in Adj(vi) | j>I } is complete. 1 8 Theorem 7 (Cont.) Definition: The following are equivalent: 1. G is chordal 2. The edges of G can be directed acyclically so that every pair of converging arrows comes from two adjacent vertices. 3. G has a perfect vertex elimination scheme. 4. There is a tree T with maximal cliques of G as vertices. Any two cliques containing v are either adjacent in T or connected by a path of cliques that contain v. 1 9 Proof of Theorem 7 (1→3) By lemma 1, G has a simplicial vertex. The subgraph induced after removing this vertex is also chordal. By induction, there is an elimination order. 2 0 Proof of Theorem 7 (3→1) Let us assume a cycle of length 4 or greater. Let x be the first vertex on the elimination scheme of this cycle. x is simplicial at this point. Two vertices adjacent to x from the cycle are also adjacent between themselves. 2 1 Theorem 7 (Cont.) Definition: The following are equivalent: 1. G is chordal 2. The edges of G can be directed acyclically so that every pair of converging arrows comes from two adjacent vertices. 3. G has a perfect vertex elimination scheme. 4. There is a tree T with maximal cliques of G as vertices. Any two cliques containing v are either adjacent in T or connected by a path of cliques that contain v. 2 2 Clique Tree Example C1 a a b c C1 b c C2 C2 b c C3 e d d C3 b e d 2 3 Proof of Theorem 7 (1→4) Lemma 2: A vertex is simplicial iff it belongs to exactly one clique. Let G =(V,E) be a chordal graph and |V|=k Proof by induction: Assume the claim holds for all graphs of size < k. By Lemma 1 G has a simplicial vertex v. By Lemma 2 v belongs to exactly one clique C. 2 4 Proof of Theorem 7 (1→4) For G’ =(V\{v},E’) there exists a tree T’ which satisfies the claim. Split into two cases: C’ = C – {v} is maximal in G’. Add v to C’ to build T from T’. T is a Clique-Tree as needed. a C 1 C C1 2 b c a b b c e d C2 e d 2 5 Proof of Theorem 7 (1→4) C’ = C – {v} is not maximal in G’. There is a maximal clique P in G’ such that C’⊂P. Add C and (C, P) to T’ to build T. T is a Clique-Tree as needed. C a b 1 C2 a C2 a b C1 c d d c d 2 6 Proof of Theorem 7 (4→1) Let T be a Clique-Tree of graph G=(V,E) and |V|=k. Proof by induction: Assume the claim holds for all graphs of size < k. Let L be a leaf in T, and P its parent in T. Let v∈L\P (exists from maximality). v cannot be in any other clique. by Lemma 2, v is simplicial. 2 7 Proof of Theorem 7 (4→1) Let T’ be T with v removed. T’ is a Clique- Tree. By assumption G’ =(V\{v},E’) is chordal. By part 3 of Theorem 7, there exists σ’, a Perfect Elimination Order for G’. Let σ = [v, σ’] be a perfect elimination order for G. By the converse of part 3, we can conclude that G is chordal 2 8 PEO is a Key in designing algorithms and obtaining structural properties The following classical Graph Optimization problems are NP-Hard for general Graphs Finding (G) :clique number (the size of maximum size clique ) (G): Chromatic number ( minimum number of colors needed in a proper coloring of G) (G): independence number( maximum size of an independent set) (G): clique covering number ( minimum number of cliques needed to cover V(G) ) All these four problems can be solved in linear time in Chordal Graphs given a PEO as part of input 29 Polynomial Algorithms for chordal Graphs: Clique Number Given a PEO of a Chordal Graph G, maximum size clique can be found as follows: Let (v1,v2,…,vn) be a PEO of G. Define L(i)= maximum j, j > i such that vivj E(G). F[i]=minimum j j >I such vivj isan edge. Let Max {L[i]-F[i]+1}=k=L[j]-F[j]+1. Then k= (G) and C={vj,vj+1,…,v_{j+k-1}} is a maximum size clique.
Load more

Recommended publications
  • A Class of Hypergraphs That Generalizes Chordal Graphs
    MATH. SCAND. 106 (2010), 50–66 A CLASS OF HYPERGRAPHS THAT GENERALIZES CHORDAL GRAPHS ERIC EMTANDER Abstract In this paper we introduce a class of hypergraphs that we call chordal. We also extend the definition of triangulated hypergraphs, given by H. T. Hà and A. Van Tuyl, so that a triangulated hypergraph, according to our definition, is a natural generalization of a chordal (rigid circuit) graph. R. Fröberg has showed that the chordal graphs corresponds to graph algebras, R/I(G), with linear resolu- tions. We extend Fröberg’s method and show that the hypergraph algebras of generalized chordal hypergraphs, a class of hypergraphs that includes the chordal hypergraphs, have linear resolutions. The definitions we give, yield a natural higher dimensional version of the well known flag property of simplicial complexes. We obtain what we call d-flag complexes. 1. Introduction and preliminaries Let X be a finite set and E ={E1,...,Es } a finite collection of non empty subsets of X . The pair H = (X, E ) is called a hypergraph. The elements of X and E , respectively, are called the vertices and the edges, respectively, of the hypergraph. If we want to specify what hypergraph we consider, we may write X (H ) and E (H ) for the vertices and edges respectively. A hypergraph is called simple if: (1) |Ei |≥2 for all i = 1,...,s and (2) Ej ⊆ Ei only if i = j. If the cardinality of X is n we often just use the set [n] ={1, 2,...,n} instead of X . Let H be a hypergraph. A subhypergraph K of H is a hypergraph such that X (K ) ⊆ X (H ), and E (K ) ⊆ E (H ).IfY ⊆ X , the induced hypergraph on Y, HY , is the subhypergraph with X (HY ) = Y and with E (HY ) consisting of the edges of H that lie entirely in Y.
    [Show full text]
  • 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.
    [Show full text]
  • On the Pathwidth of Chordal Graphs
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Discrete Applied Mathematics 45 (1993) 233-248 233 North-Holland On the pathwidth of chordal graphs Jens Gustedt* Technische Universitdt Berlin, Fachbereich Mathematik, Strape des I7 Jmi 136, 10623 Berlin, Germany Received 13 March 1990 Revised 8 February 1991 Abstract Gustedt, J., On the pathwidth of chordal graphs, Discrete Applied Mathematics 45 (1993) 233-248. In this paper we first show that the pathwidth problem for chordal graphs is NP-hard. Then we give polynomial algorithms for subclasses. One of those classes are the k-starlike graphs - a generalization of split graphs. The other class are the primitive starlike graphs a class of graphs where the intersection behavior of maximal cliques is strongly restricted. 1. Overview The pathwidth problem-PWP for short-has been studied in various fields of discrete mathematics. It asks for the size of a minimum path decomposition of a given graph, There are many other problems which have turned out to be equivalent (or nearly equivalent) formulations of our problem: - the interval graph extension problem, - the gate matrix layout problem, - the node search number problem, - the edge search number problem, see, e.g. [13,14] or [17]. The first three problems are easily seen to be reformula- tions. For the fourth there is an easy transformation to the third [14]. Section 2 in- troduces the problem as well as other problems and classes of graphs related to it. Section 3 gives basic facts on path decompositions.
    [Show full text]
  • Treewidth and Pathwidth of Permutation Graphs
    Treewidth and pathwidth of permutation graphs Citation for published version (APA): Bodlaender, H. L., Kloks, A. J. J., & Kratsch, D. (1992). Treewidth and pathwidth of permutation graphs. (Universiteit Utrecht. UU-CS, Department of Computer Science; Vol. 9230). Utrecht University. Document status and date: Published: 01/01/1992 Document Version: Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers) Please check the document version of this publication: • A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website. • The final author version and the galley proof are versions of the publication after peer review. • The final published version features the final layout of the paper including the volume, issue and page numbers. Link to publication General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal.
    [Show full text]
  • Tree-Decomposition Graph Minor Theory and Algorithmic Implications
    DIPLOMARBEIT Tree-Decomposition Graph Minor Theory and Algorithmic Implications Ausgeführt am Institut für Diskrete Mathematik und Geometrie der Technischen Universität Wien unter Anleitung von Univ.Prof. Dipl.-Ing. Dr.techn. Michael Drmota durch Miriam Heinz, B.Sc. Matrikelnummer: 0625661 Baumgartenstraße 53 1140 Wien Datum Unterschrift Preface The focus of this thesis is the concept of tree-decomposition. A tree-decomposition of a graph G is a representation of G in a tree-like structure. From this structure it is possible to deduce certain connectivity properties of G. Such information can be used to construct efficient algorithms to solve problems on G. Sometimes problems which are NP-hard in general are solvable in polynomial or even linear time when restricted to trees. Employing the tree-like structure of tree-decompositions these algorithms for trees can be adapted to graphs of bounded tree-width. This results in many important algorithmic applications of tree-decomposition. The concept of tree-decomposition also proves to be useful in the study of fundamental questions in graph theory. It was used extensively by Robertson and Seymour in their seminal work on Wagner’s conjecture. Their Graph Minors series of papers spans more than 500 pages and results in a proof of the graph minor theorem, settling Wagner’s conjecture in 2004. However, it is not only the proof of this deep and powerful theorem which merits mention. Also the concepts and tools developed for the proof have had a major impact on the field of graph theory. Tree-decomposition is one of these spin-offs. Therefore, we will study both its use in the context of graph minor theory and its several algorithmic implications.
    [Show full text]
  • Chordal Graphs MPRI 2017–2018
    Chordal graphs MPRI 2017{2018 Chordal graphs MPRI 2017{2018 Michel Habib [email protected] http://www.irif.fr/~habib Sophie Germain, septembre 2017 Chordal graphs MPRI 2017{2018 Schedule Chordal graphs Representation of chordal graphs LBFS and chordal graphs More structural insights of chordal graphs Other classical graph searches and chordal graphs Greedy colorings Chordal graphs MPRI 2017{2018 Chordal graphs Definition A graph is chordal iff it has no chordless cycle of length ≥ 4 or equivalently it has no induced cycle of length ≥ 4. I Chordal graphs are hereditary I Interval graphs are chordal Chordal graphs MPRI 2017{2018 Chordal graphs Applications I Many NP-complete problems for general graphs are polynomial for chordal graphs. I Graph theory : Treewidth (resp. pathwidth) are very important graph parameters that measure distance from a chordal graph (resp. interval graph). 1 I Perfect phylogeny 1. In fact chordal graphs were first defined in a biological modelisation pers- pective ! Chordal graphs MPRI 2017{2018 Chordal graphs G.A. Dirac, On rigid circuit graphs, Abh. Math. Sem. Univ. Hamburg, 38 (1961), pp. 71{76 Chordal graphs MPRI 2017{2018 Representation of chordal graphs About Representations I Interval graphs are chordal graphs I How can we represent chordal graphs ? I As an intersection of some family ? I This family must generalize intervals on a line Chordal graphs MPRI 2017{2018 Representation of chordal graphs Fundamental objects to play with I Maximal Cliques under inclusion I We can have exponentially many maximal cliques : A complete graph Kn in which each vertex is replaced by a pair of false twins has exactly 2n maximal cliques.
    [Show full text]
  • Distance-Hereditary and Strongly Distance-Hereditary Graphs
    Volume 88 BULLETIN of The 02 02 2020 INSTITUTE of COMBINATORICS and its APPLICATIONS Editors-in-Chief: Marco Buratti, Donald Kreher, Tran van Trung ISSN: 2689-0674 (Online) Boca Raton, FL, U.S.A. ISSN: 1183-1278 (Print) BULLETIN OF THE ICA Volume 88 (2020), Pages 50{64 Distance-hereditary and strongly distance-hereditary graphs Terry A. McKee Wright State University, Dayton, Ohio USA [email protected] Abstract: A graph G is distance-hereditary if the distance between vertices in connected induced subgraphs always equals the distance between them in G; equivalently, G contains no induced cycle of length 5 or more and no induced house, domino, or gem subgraph. Define G to be strongly distance-hereditary if G is distance-hereditary and G2 is strongly chordal. Although this definition seems completely unmo- tivated, it parallels a couple of ways in which strongly chordal graphs are the natural strengthening of chordal graphs; for instance, being distance- hereditary is characterized by the k = 1 case of a k 1 characterization of being strongly distance-hereditary. Moreover, there∀ ≥ is an induced forbid- den subgraph characterization of a distance-hereditary graph being strongly distance-hereditary. 1 Introduction and definitions A graph G is distance-hereditary if, in every connected induced subgraph G0 of G, the distance between vertices in G0 equals their distance in G; see [2,4,8]. Proposition 1.1 will give three basic characterizations, where (1) is from [8], and (2) and (3) are from [2] (also see [1]). In characterization (1), chords ab and cd are crossing chords of a cycle C if the four vertices come AMS (MOS) Subject Classifications: Primary 05C75; secondary 05C12 Key words and phrases: distance-hereditary graph; strongly chordal graph Received: 8 June 2019 50 Accepted: 23 October 2019 in the order a, c, b, d around C.
    [Show full text]
  • Chalk & Talk Session 1 Chordal Graph
    CSA E0 221: Discrete Structures November 19, 2014 Chalk & Talk Session Instructor: Arpita Patra Submitted by: Lakhpat Meena and Pankaj Gautam 1 Chordal Graph : Definition 1 A graph is called chordal if every cycle with more than three vertices has an edge connecting two non-consecutive vertices or has an edge that is not part of the cycle but connects two vertices of the cycle. Figure 1: Chordal graph Why Chordal Graphs are important ? • Chordal graphs are a subset of the perfect graphs. • They may be recognized in polynomial time, and several problems that are hard on other classes of graphs such as graph coloring may be solved in polynomial time when the input is chordal graph. • All induced subgraphs of a chordal graph are chordal. • Chordal graphs are also called triangulated and perfect elimination graphs. Definition 2 In a graph G, a vertex v is called simplicial if and only if the subgraph of G induced by the vertex set fvg [ N(v) is a complete graph. in the graph above, vertex A is simplicial, while vertex D is not. 1-1 Definition 3 A graph G on n vertices is said to have a perfect elimination ordering if and only if there is an ordering fv1; :::vng of Gs vertices, such that each vi is simplicial in the subgraph induced by the vertices fv1; :::vig. As an example, the graph above has a perfect elimination ordering, witnessed by the ordering (2, 1, 3, 4) of its vertices. Theorem 1 (Dirac, 1961) Each chordal graph has a simplicial vertex and if G is not a clique it has two non adjacent simplicial vertices.
    [Show full text]
  • Graph Theory
    Graph Theory Perfect Graphs PLANARITY 2 PERFECT GRAPHS Perfect Graphs • A perfect graph is a graph in which the chromatic number of every induced subgraph equals the size of the largest clique of that subgraph (clique number). • An arbitrary graph 퐺 is perfect if and only if we have: ∀ 푆 ⊆ 푉 퐺 휒 퐺 푆 = 휔 퐺 푆 • Theorem 1 (Perfect Graph Theorem) A graph 퐺 is perfect if and only if its complement 퐺 is perfect • Theorem 2 (Strong Perfect Graph Theorem) Perfect graphs are the same as Berge graphs, which are graphs 퐺 where neither 퐺 nor 퐺 contain an induced cycle of odd length 5 or more. 3 PERFECT GRAPHS Perfect Graphs Perfect Comparability Chordal Bipartite Permutation Split Interval Chordal Cograph Bipartite Convex Quasi- Bipartite Thresshold Biconvex Thresshold Bipartite Bipartite 4 Permutation PERFECT GRAPHS Intersection Graphs • Let 퐹 be a family of nonempty sets. The intersection graph of 퐹 is obtained be representing each set in 퐹 by a vertex: 푥 → 푦 ⇔ 푆푋∩푆푌 ≠ ∅ • The intersection graph of a family of intervals on a linearly ordered set (like the real line) is called an Interval graph. • An induced subgraph of an interval graph is an interval graph. 5 PERFECT GRAPHS Intersection Graphs • Let 퐹 be a family of nonempty sets. The intersection graph of 퐹 is obtained be representing each set in 퐹 by a vertex: 푥 → 푦 ⇔ 푆푋∩푆푌 ≠ ∅ • Circular-arc graphs properly contain the internal graphs. 6 PERFECT GRAPHS Intersection Graphs • Let 퐹 be a family of nonempty sets. The intersection graph of 퐹 is obtained be representing each set in 퐹 by a vertex: 푥 → 푦 ⇔ 푆푋∩푆푌 ≠ ∅ • A permutation diagram consists of n points on each of two parallel lines and n straight line segments matching the points.
    [Show full text]
  • Graph Searching & Perfect Graphs
    Graph Searching & Perfect Graphs∗ Lalla Mouatadid University of Toronto Abstract Perfect graphs, by definition, have a nice structure, that graph searching seems to extract in a, often non-inexpensive, manner. We scratch the surface of this elegant research area by giving two examples: Lexicographic Breadth Search on Chordal Graphs, and Lexicographic Depth First Search on Cocomparability graphs. 1 Introduction Here's one particular way to cope with NP-hardness: Suppose we know some structure about the object we are dealing with, can we exploit said structure to come up with simple (hmm...?), efficient algorithms to NP-hard problems. One example of this structure is the \interval representation" most people are familiar with, where in such a setting we were able to solve the independent set problem in linear time, whereas the independent set problem on arbitrary graphs cannot even be approximated in polynomial time to a constant factor (unless P = NP). Before we delve into the topic, let's recall some definitions. Let G(V; E) be a finite and simple (no loops and no multiple edges) graph: • The neighbourhood of a vertex v is the set of vertices adjacent to v. We write N(v) = fujuv 2 Eg. The closed neighbourhood of v is N[v] = N(v) [ fvg. • The complement of G, denoted G(V; E) is the graph on the same set of vertices as G, where for all u; v 2 V; uv 2 E () uv2 = E. • A induced subgraph of G is a graph G0(V 0;E0) where V 0 ⊆ V and 8u; v 2 V 0, uv 2 E0 () uv 2 E.
    [Show full text]
  • Perfect Graphs Chinh T
    University of Dayton eCommons Computer Science Faculty Publications Department of Computer Science 2015 Perfect Graphs Chinh T. Hoang Wilfrid Laurier University R. Sritharan University of Dayton Follow this and additional works at: http://ecommons.udayton.edu/cps_fac_pub Part of the Graphics and Human Computer Interfaces Commons, and the Other Computer Sciences Commons eCommons Citation Hoang, Chinh T. and Sritharan, R., "Perfect Graphs" (2015). Computer Science Faculty Publications. 87. http://ecommons.udayton.edu/cps_fac_pub/87 This Book Chapter is brought to you for free and open access by the Department of Computer Science at eCommons. It has been accepted for inclusion in Computer Science Faculty Publications by an authorized administrator of eCommons. For more information, please contact [email protected], [email protected]. CHAPTE R 28 Perfect Graphs Chinh T. Hoang* R. Sritharan t CO NTENTS 28.1 Introd uction ... .. ..... ............. ............................... .. ........ .. 708 28.2 Notation ............................. .. ..................... .................... 710 28.3 Chordal Graphs ....................... ................. ....... ............. 710 28.3.1 Characterization ............ ........................... .... .. ... 710 28.3.2 Recognition .................... ..................... ... .. ........ .. ... 712 28.3.3 Optimization ................................................ .. ......... 715 28.4 Comparability Graphs ............................................... ..... .. 715 28.4.1 Characterization
    [Show full text]
  • A Cubic Vertex-Kernel for Trivially Perfect Editing
    A cubic vertex-kernel for Trivially Perfect Editing Ma¨elDumas1, Anthony Perez1, and Ioan Todinca1 1Univ. Orl´eans,INSA Centre Val de Loire, LIFO EA 4022, F-45067 Orl´eans,France May 19, 2021 Abstract We consider the Trivially Perfect Editing problem, where one is given an undirected graph G = (V; E) and a parameter k 2 N and seeks to edit (add or delete) at most k edges from G to obtain a trivially perfect graph. The related Trivially Perfect Completion and Trivially Perfect Deletion problems are obtained by only allowing edge additions or edge deletions, respectively. Trivially perfect graphs are both chordal and cographs, and have applications related to the tree- depth width parameter and to social network analysis. All variants of the problem are known to be NP-Complete [6, 28] and to admit so-called polynomial kernels [13, 22]. More precisely, the existence of an O(k3) vertex-kernel for Trivially Perfect Completion was announced by Guo [22] but without a stand-alone proof. More recently, Drange and Pilipczuk [13] provided O(k7) vertex- kernels for these problems and left open the existence of cubic vertex-kernels. In this work, we answer positively to this question for all three variants of the problem. Introduction A broad range of optimization problems on graphs are particular cases of so-called modification problems. Given an arbitrary graph G = (V; E) and an integer k, the question is whether G can be turned into a graph satisfying some desired property by at most k modifications. By modifications we mean, according to the problem, vertex deletions (as for Vertex Cover and Feedback Vertex Set where we aim to obtain graphs with no edges, or without cycles respectively) or edge deletions and/or additions (as for Minimum Fill-In, also known as Chordal Completion, where the goal is to obtain a chordal graph, with no induced cycles with four or more vertices, by adding at most k edges).
    [Show full text]