DOCSLIB.ORG

Ssp Structure of Some Graph Classes

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Ssp Structure of Some Graph Classes International Journal of Pure and Applied Mathematics Volume 101 No. 6 2015, 939-948 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: A http://www.ijpam.eu P ijpam.eu SSP STRUCTURE OF SOME GRAPH CLASSES R. Mary Jeya Jothi1, A. Amutha2 1,2Department of Mathematics Sathyabama University Chennai, 119, INDIA Abstract: A Graph G is Super Strongly Perfect if every induced sub graph H of G possesses a minimal dominating set that meets all maximal cliques of H. Strongly perfect, complete bipartite graph, etc., are some of the most important classes of super strongly perfect graphs. Here, we analyse the other classes of super strongly perfect graphs like trivially perfect graphs and very strongly perfect graphs. We investigate the structure of super strongly perfect graph in friendship graphs. Also, we discuss the maximal cliques, colourability and dominating sets in friendship graphs. AMS Subject Classification: 05C75 Key Words: super strongly perfect graph, trivially perfect graph, very strongly perfect graph and friendship graph 1. Introduction Super strongly perfect graph is defined by B.D. Acharya and its characterization has been given as an open problem in 2006 [5]. Many analysation on super strongly perfect graph have been done by us in our previous research work, (i.e.,) we have investigated the classes of super strongly perfect graphs like strongly c 2015 Academic Publications, Ltd. Received: March 12, 2015 url: www.acadpubl.eu 940 R.M.J. Jothi, A. Amutha perfect, complete bipartite graph, etc., [3, 4]. In this paper we have discussed some other classes of super strongly perfect graphs like trivially perfect graph, very strongly perfect graph and friendship graph. Also, we analyse the maximal cliques, colourability and minimal dominating sets in friendship graph. In this paper, graphs are finite, undirected and simple, that is, they have no loops or multiple edges. Let G be a graph. A Clique in G is a set X ⊆ V (G) of pair wise adjacent vertices. A subset D of V (G) is called a Dominating set if every vertex in V \D is adjacent to at least one vertex in D. A subset S of V is said to be a Minimal Dominating Set if S − {u} is not a dominating set for any u ∈ S. By vertex colouring or colouring of a graph G, we mean assigning different colours to the vertices of the graph such that no two adjacent vertices get the same colour. The Chromatic number of a graph G is the minimum number of colours that is required to colour G. A graph G = (V, E) is Strongly Perfect (SP) if every induced sub graphs H of G contains a stable set that meets all the maximal cliques of H. 2. Super Strongly Perfect Graph (SSP) A Graph G = (V, E) is super strongly perfect if every induced sub graph H of G possesses a minimal dominating set that meets all maximal cliques of H. The following Figures 1 and 2 illustrate the structure of the super strongly perfect and non super strongly perfect graphs. Example 1. Figure 1: Super strongly perfect graph Here, {3, 6} is a minimal dominating set which meets all maximal cliques of G. Example 2. Here, {1, 3, 6, 8} is a minimal dominating set which does not meet all max- imal cliques of G. SSPSTRUCTUREOFSOMEGRAPHCLASSES 941 Figure 2: Non-super strongly perfect graph Theorem 3. [4] Every strongly perfect graph is super strongly perfect. Theorem 4. [3] Let G be a graph with at least one maximal clique Kn, n = 2, 3,... then G is super strongly perfect if and only if it is n-colourable. 3. Helm Graph The Helm graph Hn is the graph obtained from an n-wheel graph by adjoining a pendant edge at each vertex of the cycle that is helm graph Hn is obtained by attaching a single edge and vertex to each vertex of the outer cycle of a wheel graph Wn. Figure 3 illustrates the helm graph H7 which is Super Strongly Perfect. Example 5. Figure 3: H7 Theorem 6. Every helm graph Hn where n is odd, is Super Strongly Perfect. 942 R.M.J. Jothi, A. Amutha Proof. Let G be a helm graph Hn where n is odd. ⇒ G is constructed by an n-wheel graph by joining a pendent edge at each vertex of the outer cycle. Since every odd wheel graph is Super Strongly Perfect, if we join a pendent edge at each vertex of the cycle of the odd wheel then again the graph is Super Strongly Perfect with a minimal dominating set of cardinality n vertices (i.e.,) the n − 1 non adjacent vertices from the n − 1 pendent edges and a middle vertex from the wheel graph. Hence G is Super Strongly Perfect. Theorem 7. Every helm graph Hn where n is even, is non-Super Strongly Perfect. Proof. Let G be a helm graph Hn where n is even. ⇒ G is constructed by an n-wheel graph by joining a pendent edge at each vertex of the cycle. Since every even wheel graph is non-Super Strongly Perfect, if we join a pendent edge at each vertex of the cycle of the even wheel, then again the graph is non-Super Strongly Perfect. Hence G is non-Super Strongly Perfect. Theorem 8. Let G be a helm graph with odd number of vertices, which is Super Strongly Perfect, then G contains (n − 1) maximal cliques K3. Proof. Let G be a helm graph with odd number of vertices, which is Super Strongly Perfect. ⇒ G is obtained from an n-wheel graph by adjoining a pendant edge at each vertex of the outer cycle Cn−1. ⇒ Any two adjacent vertices from Cn−1 with a centre single vertex give an induced K3. Since G has n − 1 such vertices, G has (n − 1) maximal cliques K3. Hence proved. Theorem 9. Let G be a helm graph with odd number of vertices, which is Super Strongly Perfect, then G is 3-colourable. Proof. Let G be a helm graph with odd number of vertices, which is Super Strongly Perfect. ⇒ by the previous theorem, G has (n − 1) maximal cliques on 3 vertices. SSPSTRUCTUREOFSOMEGRAPHCLASSES 943 ⇒ by the Theorem 4, G is 3-colourable. Hence proved. Theorem 10. Let G be a helm graph with odd number of vertices, which is Super Strongly Perfect, then G contains a minimal dominating set of cardinality n where n is the number of vertices. Proof. Let G be a Helm graph with odd number of vertices, which is Super Strongly Perfect. ⇒ G is constructed by an n-wheel graph by joining a pendent edge at each vertex of the outer cycle. ⇒ The n−1 non adjacent pendent edges of the vertices and a middle vertex from the wheel graph gives a minimal dominating set which meet all maximal cliques K3. ⇒ G has a minimal dominating set of cardinality n. Hence proved. 4. Friendship Graph A Friendship graph Fk on 2k+1 vertices is the graph obtained by taking k copies of the cycle graph C3 with a vertex in common, that is, the graph of k triangles intersecting in a single vertex. Hence every friendship graph contains a single vertex which is adjacent to all the remaining vertices. It is unit distance with girth 3, diameter 2 and radius 1. The friendship theorem of Paul Erdos, et.al. [1], states that the finite graphs with the property that every two vertices have exactly one neighbour in common are exactly the friendship graphs. Informally, if a group of people has the property that every pair of people has exactly a friend in common, then there must be one person who is a friend to all the others. However, for infinite graphs, there can be many different graphs with the same cardinality that have this property. Figure 4 illustrates the friendship graph F6 which is super strongly perfect. Example 11. Here {1} is a minimal dominating set which meets all the maximal cliques of G. Theorem 12. Every friendship graph is super strongly perfect. 944 R.M.J. Jothi, A. Amutha Figure 4: F6 Proof. Let G be a friendship graph. ⇒ Let S = {v} be the minimal dominating set of cardinality 1. ⇒ v is adjacent to all the remaining vertices of G. ⇒ v meets all maximal cliques of G. ⇒ G is super strongly perfect. Hence proved. Observation 13. Let G be a friendship graph Fk. Then G is 3-colourable. Proof. Let G be a friendship graph Fk. ⇒ G has k maximal cliques on 3 vertices. By the Theorem 4, G is 3-colourable. Hence proved. Observation 14. Let G be a friendship graph Fk. Then G has a minimal dominating set of maximum cardinality k. Proof. Let G be a friendship graph Fk. ⇒ G has k maximal cliques on 3 vertices. Since G has k copies of K3, there exists a minimal dominating set of cardi- nality k which meet all K3. ⇒ G has a minimal dominating set of maximum cardinality k. Hence proved. 5. Trivially Perfect Graph (TP) A Graph G is trivially perfect if in every induced sub graph H of G the size of the largest independent set equals the number of maximal cliques. Figure 5 SSPSTRUCTUREOFSOMEGRAPHCLASSES 945 illustrates the trivially perfect graph which is super strongly perfect. Example 15. Figure 5: Trivially perfect graph Here, the cardinality of the largest independent set is 2 and also the graph has 2 maximal cliques of size 4. Theorem 16. Every trivially perfect graph is strongly perfect. Proof. Let G be a trivially perfect graph.
Load more

Recommended publications
  • Arxiv:2106.16130V1 [Math.CO] 30 Jun 2021 in the Special Case of Cyclohedra, and by Cardinal, Langerman and P´Erez-Lantero [5] in the Special Case of Tree Associahedra
    LAGOS 2021 Bounds on the Diameter of Graph Associahedra Jean Cardinal1;4 Universit´elibre de Bruxelles (ULB) Lionel Pournin2;4 Mario Valencia-Pabon3;4 LIPN, Universit´eSorbonne Paris Nord Abstract Graph associahedra are generalized permutohedra arising as special cases of nestohedra and hypergraphic polytopes. The graph associahedron of a graph G encodes the combinatorics of search trees on G, defined recursively by a root r together with search trees on each of the connected components of G − r. In particular, the skeleton of the graph associahedron is the rotation graph of those search trees. We investigate the diameter of graph associahedra as a function of some graph parameters. It is known that the diameter of the associahedra of paths of length n, the classical associahedra, is 2n − 6 for a large enough n. We give a tight bound of Θ(m) on the diameter of trivially perfect graph associahedra on m edges. We consider the maximum diameter of associahedra of graphs on n vertices and of given tree-depth, treewidth, or pathwidth, and give lower and upper bounds as a function of these parameters. Finally, we prove that the maximum diameter of associahedra of graphs of pathwidth two is Θ(n log n). Keywords: generalized permutohedra, graph associahedra, tree-depth, treewidth, pathwidth 1 Introduction The vertices and edges of a polyhedron form a graph whose diameter (often referred to as the diameter of the polyhedron for short) is related to a number of computational problems. For instance, the question of how large the diameter of a polyhedron arises naturally from the study of linear programming and the simplex algorithm (see, for instance [27] and references therein).
    [Show full text]
  • Vertex Deletion Problems on Chordal Graphs∗†
    Vertex Deletion Problems on Chordal Graphs∗† Yixin Cao1, Yuping Ke2, Yota Otachi3, and Jie You4 1 Department of Computing, Hong Kong Polytechnic University, Hong Kong, China [email protected] 2 Department of Computing, Hong Kong Polytechnic University, Hong Kong, China [email protected] 3 Faculty of Advanced Science and Technology, Kumamoto University, Kumamoto, Japan [email protected] 4 School of Information Science and Engineering, Central South University and Department of Computing, Hong Kong Polytechnic University, Hong Kong, China [email protected] Abstract Containing many classic optimization problems, the family of vertex deletion problems has an important position in algorithm and complexity study. The celebrated result of Lewis and Yan- nakakis gives a complete dichotomy of their complexity. It however has nothing to say about the case when the input graph is also special. This paper initiates a systematic study of vertex deletion problems from one subclass of chordal graphs to another. We give polynomial-time algorithms or proofs of NP-completeness for most of the problems. In particular, we show that the vertex deletion problem from chordal graphs to interval graphs is NP-complete. 1998 ACM Subject Classification F.2.2 Analysis of Algorithms and Problem Complexity, G.2.2 Graph Theory Keywords and phrases vertex deletion problem, maximum subgraph, chordal graph, (unit) in- terval graph, split graph, hereditary property, NP-complete, polynomial-time algorithm Digital Object Identifier 10.4230/LIPIcs.FSTTCS.2017.22 1 Introduction Generally speaking, a vertex deletion problem asks to transform an input graph to a graph in a certain class by deleting a minimum number of vertices.
    [Show full text]
  • The Hadwiger Number, Chordal Graphs and Ab-Perfection Arxiv
    The Hadwiger number, chordal graphs and ab-perfection∗ Christian Rubio-Montiel [email protected] Instituto de Matem´aticas, Universidad Nacional Aut´onomade M´exico, 04510, Mexico City, Mexico Department of Algebra, Comenius University, 84248, Bratislava, Slovakia October 2, 2018 Abstract A graph is chordal if every induced cycle has three vertices. The Hadwiger number is the order of the largest complete minor of a graph. We characterize the chordal graphs in terms of the Hadwiger number and we also characterize the families of graphs such that for each induced subgraph H, (1) the Hadwiger number of H is equal to the maximum clique order of H, (2) the Hadwiger number of H is equal to the achromatic number of H, (3) the b-chromatic number is equal to the pseudoachromatic number, (4) the pseudo-b-chromatic number is equal to the pseudoachromatic number, (5) the arXiv:1701.08417v1 [math.CO] 29 Jan 2017 Hadwiger number of H is equal to the Grundy number of H, and (6) the b-chromatic number is equal to the pseudo-Grundy number. Keywords: Complete colorings, perfect graphs, forbidden graphs characterization. 2010 Mathematics Subject Classification: 05C17; 05C15; 05C83. ∗Research partially supported by CONACyT-Mexico, Grants 178395, 166306; PAPIIT-Mexico, Grant IN104915; a Postdoctoral fellowship of CONACyT-Mexico; and the National scholarship programme of the Slovak republic. 1 1 Introduction Let G be a finite graph. A k-coloring of G is a surjective function & that assigns a number from the set [k] := 1; : : : ; k to each vertex of G.A k-coloring & of G is called proper if any two adjacent verticesf haveg different colors, and & is called complete if for each pair of different colors i; j [k] there exists an edge xy E(G) such that x &−1(i) and y &−1(j).
    [Show full text]
  • Graphic Sequences with a Realization Containing a Generalized Friendship Graph ∗
    Graphic Sequences with a Realization Containing a Generalized Friendship Graph ∗ Jian-Hua Yina,† Gang Chenb, John R. Schmittc aDepartment of Applied Mathematics, College of Information Science and Technology, Hainan University, Haikou 570228, P.R. China bDepartment of Mathematics, Ningxia University, Yinchuan 750021, P.R. China cDepartment of Mathematics, Middlebury College, Middlebury, VT, USA Abstract: Gould, Jacobson and Lehel (Combinatorics, Graph Theory and Algorithms, Vol.I (1999) 451–460) considered a variation of the classical Tur´an-type extremal problems as follows: for any simple graph H, determine the smallest even integer σ(H, n) such that every n-term graphic sequence π = (d1, d2, . , dn) with term sum σ(π) = d1 + d2 + ··· + dn ≥ σ(H, n) has a realization G containing H as a subgraph. Let Ft,r,k denote the generalized friendship graph on kt − kr + r vertices, that is, the graph of k copies of Kt meeting in a common r set, where Kt is the complete graph on t vertices and 0 ≤ r ≤ t. In this paper, we determine σ(Ft,r,k, n) for k ≥ 2, t ≥ 3, 1 ≤ r ≤ t − 2 and n sufficiently large. Keywords: degree sequence, potentially Ft,r,k-graphic sequence, generalized friendship graph. 2000 MR Subject Classification: 05C35, 05C07. 1. Introduction The set of all sequences π = (d1, d2, . , dn) of non-negative, non-increasing integers with d1 ≤ n − 1 is denoted by NSn. A sequence π ∈ NSn is said to be graphic if it is the degree sequence of a simple graph G on n vertices, and such a graph G is called a realization of π.
    [Show full text]
  • On Dasgupta's Hierarchical Clustering Objective and Its Relation to Other
    On Dasgupta's hierarchical clustering objective and its relation to other graph parameters Svein Høgemo1, Benjamin Bergougnoux1, Ulrik Brandes3, Christophe Paul2, and Jan Arne Telle1 1 Department of Informatics, University of Bergen, Norway 2 LIRMM, CNRS, Univ Montpellier, France 3 Social Networks Lab, ETH Z¨urich, Switzerland Abstract. The minimum height of vertex and edge partition trees are well-studied graph parameters known as, for instance, vertex and edge ranking number. While they are NP-hard to determine in general, linear- time algorithms exist for trees. Motivated by a correspondence with Das- gupta's objective for hierarchical clustering we consider the total rather than maximum depth of vertices as an alternative objective for mini- mization. For vertex partition trees this leads to a new parameter with a natural interpretation as a measure of robustness against vertex removal. As tools for the study of this family of parameters we show that they have similar recursive expressions and prove a binary tree rotation lemma. The new parameter is related to trivially perfect graph completion and there- fore intractable like the other three are known to be. We give polynomial- time algorithms for both total-depth variants on caterpillars and on trees with a bounded number of leaf neighbors. For general trees, we obtain a 2-approximation algorithm. 1 Introduction Clustering is a central problem in data mining and statistics. Although many objective functions have been proposed for (flat) partitions into clusters, hier- archical clustering has long been considered from the perspective of iterated merge (in agglomerative clustering) or split (in divisive clustering) operations.
    [Show full text]
  • PDF of the Phd Thesis
    Durham E-Theses Topics in Graph Theory: Extremal Intersecting Systems, Perfect Graphs, and Bireexive Graphs THOMAS, DANIEL,JAMES,RHYS How to cite: THOMAS, DANIEL,JAMES,RHYS (2020) Topics in Graph Theory: Extremal Intersecting Systems, Perfect Graphs, and Bireexive Graphs , Durham theses, Durham University. Available at Durham E-Theses Online: http://etheses.dur.ac.uk/13671/ Use policy The full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that: • a full bibliographic reference is made to the original source • a link is made to the metadata record in Durham E-Theses • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders. Please consult the full Durham E-Theses policy for further details. Academic Support Oce, Durham University, University Oce, Old Elvet, Durham DH1 3HP e-mail: [email protected] Tel: +44 0191 334 6107 http://etheses.dur.ac.uk 2 Topics in Graph Theory: Extremal Intersecting Systems, Perfect Graphs, and Bireflexive Graphs Daniel Thomas A Thesis presented for the degree of Doctor of Philosophy Department of Computer Science Durham University United Kingdom June 2020 Topics in Graph Theory: Extremal Intersecting Systems, Perfect Graphs, and Bireflexive Graphs Daniel Thomas Submitted for the degree of Doctor of Philosophy June 2020 Abstract: In this thesis we investigate three different aspects of graph theory. Firstly, we consider interesecting systems of independent sets in graphs, and the extension of the classical theorem of Erdős, Ko and Rado to graphs.
    [Show full text]
  • Math.RA] 25 Sep 2013 Previous Paper [3], Also Relying in Conceptually Separated Tools from Them, Such As Graphs and Digraphs
    Certain particular families of graphicable algebras Juan Núñez, María Luisa Rodríguez-Arévalo and María Trinidad Villar Dpto. Geometría y Topología. Facultad de Matemáticas. Universidad de Sevilla. Apdo. 1160. 41080-Sevilla, Spain. [email protected] [email protected] [email protected] Abstract In this paper, we introduce some particular families of graphicable algebras obtained by following a relatively new line of research, ini- tiated previously by some of the authors. It consists of the use of certain objects of Discrete Mathematics, mainly graphs and digraphs, to facilitate the study of graphicable algebras, which are a subset of evolution algebras. 2010 Mathematics Subject Classification: 17D99; 05C20; 05C50. Keywords: Graphicable algebras; evolution algebras; graphs. Introduction The main goal of this paper is to advance in the research of a novel mathematical topic emerged not long ago, the evolution algebras in general, and the graphicable algebras (a subset of them) in particular, in order to obtain new results starting from those by Tian (see [4, 5]) and others already obtained by some of us in a arXiv:1309.6469v1 [math.RA] 25 Sep 2013 previous paper [3], also relying in conceptually separated tools from them, such as graphs and digraphs. Concretely, our goal is to find some particular types of graphicable algebras associated with well-known types of graphs. The motivation to deal with evolution algebras in general and graphicable al- gebras in particular is due to the fact that at present, the study of these algebras is very booming, due to the numerous connections between them and many other branches of Mathematics, such as Graph Theory, Group Theory, Markov pro- cesses, dynamic systems and the Theory of Knots, among others.
    [Show full text]
  • On Retracts, Absolute Retracts, and Folds In
    ON RETRACTS,ABSOLUTE RETRACTS, AND FOLDS IN COGRAPHS Ton Kloks1 and Yue-Li Wang2 1 Department of Computer Science National Tsing Hua University, Taiwan [email protected] 2 Department of Information Management National Taiwan University of Science and Technology [email protected] Abstract. Let G and H be two cographs. We show that the problem to determine whether H is a retract of G is NP-complete. We show that this problem is fixed-parameter tractable when parameterized by the size of H. When restricted to the class of threshold graphs or to the class of trivially perfect graphs, the problem becomes tractable in polynomial time. The problem is also solvable in linear time when one cograph is given as an in- duced subgraph of the other. We characterize absolute retracts for the class of cographs. Foldings generalize retractions. We show that the problem to fold a trivially perfect graph onto a largest possible clique is NP-complete. For a threshold graph this folding number equals its chromatic number and achromatic number. 1 Introduction Graph homomorphisms have regained a lot of interest by the recent characteri- zation of Grohe of the classes of graphs for which Hom(G, -) is tractable [11]. To be precise, Grohe proves that, unless FPT = W[1], deciding whether there is a homomorphism from a graph G 2 G to some arbitrary graph H is polynomial if and only if the graphs in G have bounded treewidth modulo homomorphic equiv- alence. The treewidth of a graph modulo homomorphic equivalence is defined as the treewidth of its core, ie, a minimal retract.
    [Show full text]
  • Interlace Polynomials of Friendship Graphs
    Electronic Journal of Graph Theory and Applications 6 (2) (2018), 269–281 Interlace polynomials of friendship graphs Christina Eubanks-Turnera, Aihua Lib aDepartment of Mathematics, Loyola Marymount University, 90045 bDepartment of Mathematical Sciences, Montclair State University, 07043 [email protected], [email protected] Abstract In this paper, we study the interlace polynomials of friendship graphs, that is, graphs that satisfy the Friendship Theorem given by Erdos,¨ Renyi´ and Sos. Explicit formulas, special values, and behaviour of coefficients of these polynomials are provided. We also give the interlace polynomials of other similar graphs, such as, the butterfly graph. Keywords: graph polynomial, interlace polynomial, friendship graph, butterfly graph Mathematics Subject Classification : 05C31, 05C50 DOI: 10.5614/ejgta.2018.6.2.7 1. Introduction Sequencing by hybridization is a method of reconstructing a long DNA string from its nu- cleotide sequence. Since gaining a unique reconstruction from the substrings is not always possi- ble, a major question that arises in this study is “For a random string, how many reconstructions are possible?” In [2], Arratia, Bollobas,´ Coppersmith, and Sorkin answer an important question related to DNA sequencing by converting this to a question about Euler circuits in a 2-in, 2-out graph that have been “toggled” (interlaced). The previously mentioned authors introduced the in- terlace polynomial of a graph, a polynomial that represents the information gained from doing the toggling process on the graph, see [2]. Interlace polynomials are similar to other graph polynomi- als, such as, Tutte and Martin polynomials, see [5]. Some researchers have studied different types of graph polynomials, such as genus polynomials, [7].
    [Show full text]
  • Linear Separation of Connected Dominating Sets in Graphs (Extended Abstract) Nina Chiarelli1 and Martin Milanicˇ2
    Linear Separation of Connected Dominating Sets in Graphs (Extended Abstract) Nina Chiarelli1 and Martin Milanicˇ2 1 University of Primorska, UP FAMNIT, Glagoljaskaˇ 8, SI6000 Koper, Slovenia [email protected] 2 University of Primorska, UP IAM, Muzejski trg 2, SI6000 Koper, Slovenia University of Primorska, UP FAMNIT, Glagoljaskaˇ 8, SI6000 Koper, Slovenia [email protected] Abstract are looking for, such as matchings, cliques, stable sets, dominating sets, etc. A connected dominating set in a graph is a dominating set of vertices that induces a The above framework provides a unified way connected subgraph. We introduce and study of describing characteristic properties of several the class of connected-domishold graphs, graph classes studied in the literature, such as which are graphs that admit non-negative real threshold graphs (Chvatal´ and Hammer 1977), weights associated to their vertices such that domishold graphs (Benzaken and Hammer 1978), a set of vertices is a connected dominating total domishold graphs (Chiarelli and Milanicˇ set if and only if the sum of the correspond- 2013a; 2013b) and equistable graphs (Payan 1980; ing weights exceeds a certain threshold. We show that these graphs form a non-hereditary Mahadev, Peled, and Sun 1994). If weights as class of graphs properly containing two above exist and are given with the graph, and the well known classes of chordal graphs: block set T is given by a membership oracle, then a dy- graphs and trivially perfect graphs. We char- namic programming algorithm can be
    [Show full text]
  • On Characterizing Game-Perfect Graphs by Forbidden Induced Subgraphs
    Volume 7, Number 1, Pages 21{34 ISSN 1715-0868 ON CHARACTERIZING GAME-PERFECT GRAPHS BY FORBIDDEN INDUCED SUBGRAPHS STEPHAN DOMINIQUE ANDRES Abstract. A graph G is called g-perfect if, for any induced subgraph H of G, the game chromatic number of H equals the clique number of H. A graph G is called g-col-perfect if, for any induced subgraph H of G, the game coloring number of H equals the clique number of H. In this paper we characterize the classes of g-perfect resp. g-col-perfect graphs by a set of forbidden induced subgraphs. Moreover, we study similar notions for variants of the game chromatic number, namely B-perfect and [A; B]-perfect graphs, and for several variants of the game coloring number, and characterize the classes of these graphs. 1. Introduction A well-known maker-breaker game is one of Bodlaender's graph coloring games [9]. We are given an initially uncolored graph G and a color set C. Two players, Alice and Bob, move alternately with Alice beginning. A move consists in coloring an uncolored vertex with a color from C in such a way that adjacent vertices receive distinct colors. The game ends if no move is possible any more. The maker Alice wins if the vertices of the graph are completely colored, otherwise, i.e. if there is an uncolored vertex surrounded by colored vertices of each color, the breaker Bob wins. For a graph G, the game chromatic number χg(G) of G is the smallest cardinality of a color set C such that Alice has a winning strategy in the game described above.
    [Show full text]
  • A Polynomial Kernel for Trivially Perfect Editing∗
    A Polynomial Kernel for Trivially Perfect Editing∗ P˚alGrøn˚asDrangey Micha lPilipczukz October 9, 2018 Abstract We give a kernel with O(k7) vertices for Trivially Perfect Editing, the problem of adding or removing at most k edges in order to make a given graph trivially perfect. This answers in affirmative an open question posed by Nastos and Gao [27], and by Liu et al. [24]. Our general technique implies also the existence of kernels of the same size for related Trivially Perfect Completion and Trivially Perfect Deletion problems. Whereas for the former an O(k3) kernel was given by Guo [19], for the latter no polynomial kernel was known. We complement our study of Trivially Perfect Editing by proving that, contrary to Trivially Perfect Completion, it cannot be solved in time 2o(k)·nO(1) unless the Exponential Time Hypothesis fails. In this manner we complete the picture of the parameterized and kernelization complexity of the classic edge modification problems for the class of trivially perfect graphs. 1 Introduction Graph modification problems form an important subclass of discrete computational problems, where the task is to modify a given graph using a constrained number of modifications in order to make it satisfy some property Π, or equivalently belong to the class G of graphs satisfying Π. Well-known examples of graph modification problems include Vertex Cover, Cluster Editing, Feedback Vertex Set, Odd Cycle Transversal, and Minimum Fill-In. The systematic study of graph modification problems dates back to early 80s and the work of Yannakakis [28], who showed that there is a dichotomy for the vertex deletion problems: unless a graph class G is trivial (finite or co-finite), the problem of deleting the least number of vertices to obtain a graph from G is NP-hard.
    [Show full text]