DOCSLIB.ORG

A Graph G Is Trivially Perfect If for Each Induced Subgraph H of G, the Number of Maximal Cliques of H Is Equal to the Maximum Size of an Independent Set of H

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
A Graph G Is Trivially Perfect If for Each Induced Subgraph H of G, the Number of Maximal Cliques of H Is Equal to the Maximum Size of an Independent Set of H Stud.Cercet.Ştiint., Ser.Mat., 17 (2007), Supplement Proceedings of CNMI 2007, Bacău, Nov. 2007, pp. 255-260 A RECOGNITION ALGORITHM FOR TRIVIALLY PERFECT GRAPHS MIHAI TALMACIU, ELENA NECHITA Abstract: A graph G is trivially perfect if for each induced subgraph H of G, the number of maximal cliques of H is equal to the maximum size of an independent set of H. In this paper we give an algorithm for trivially perfect graphs. Keywords and phrases:Trivially perfect graphs, weakly decomposition,recognition algorithm. (2000) Mathematics Subject Classification: 54 1 Introduction Let G=(V,E) be a graph with n vertices and m edges. Our algorithm rely on a new characterization of trivially perfect graphs. The paper is organized as follows. In Section 2 we give notations and definitions. For the unity of the paper, in Section 3 we shortly remind the weakly decomposition [5]. In Section 4 we present the necessary and sufficient conditions for a graph to be an trivially perfect graphs, in order to determine the number clique and the stability number. 2 Notations and definitions Throughout this paper [1] G=(V,E) is a simple (i.e. finite, undirected, without loops and multiple edges) graph. Let co-GG= denote the complement graph of G. For U ⊆ V let [U] denote the subgraph of G induced by U. By G-X we mean the graph [V-X], whenever X⊆ V, but we shall often denote it simply G- v (∀ v∈V) when there is no ambiguity. If v∈V is a vertex in G, the neighborhood NvG ( ) denotes the vertices of G-v that are adjacent to v. We shall write N(v) when the graph G appears clearly from the context. The neighborhood of the vertex v in the complement of the graph G will be 255 denoted by Nv(). For any subset S of vertices in the graph G the neighborhood of S is NS( )= ∪−vS∈ Nv ( ) S and N[S]=S ∪ N(S). A clique is a subset of V with the property that all the vertices are pairwise adjacent. The clique number of G, denoted by ω (G) is the size of the maximum clique. By Pn, Cn, Kn we mean a chordless path on n ≥ 3 vertices, the chordless cycle on n ≥ 3 vertices, and the complete graph on n ≥ 1 vertices. If e=xy∈E, we shall also write x ∼ y, and x ≠ y whenever x, y are not adjacent in G. A set A is totally adjacent (non adjacent) with a set B of vertices (A ∩ B=φ ) if ab is (is not) edge, for any a vertex in A and any b vertex in B; we note with A ∼ B (A ≠ B). A graph G is F-free if none of its induced subgraphs is in F. 3 The weakly decomposition Here we recall the notions and the results (see [3], also [5]) that are necessary in the next section. For this we define the notion of weakly component and give a characterization for the weakly decomposition of a graph. Definition 1. [3], also [5]) Let G=(V,E) be a graph. A set of vertices, A, is called weakly set if NG(A)=V-A and the induced subgraph by A is connected. If A is a weakly set, maximal with respect to the inclusion, the subgraph induced by A is called weakly component. For simplification, the weakly component G(A) will be denoted with A. The name of "weakly component" is justified by the next result. Theorem 1. ([3], also [5]) Any connected and incomplete graph G=(V,E) admits a weakly component A such that GV(−= A ) G (())(()) N A + G N A . Theorem 2. ([3], also [5]) Let G=(V,E) be a connected and incomplete graph and A ⊂ V. Then A is a weakly component of G if and only if G(A) is connected and N(A)~N(A) . Definition 2. ([3], also [5]) A partition (A,N(A),V-A ∪ N(A)), where A is a weakly set, is called weakly decomposition of graph G in relation to A. We call: A the weakly component, N(A) the minimal cutset, and V-N(A) the remote set. The next result insures the existence of a weakly decomposition in a connected and incomplete graph. Theorem 3. ([3], also [5]) If G=(V,E) is a connected and incomplete graph then the set of vertices V admits a weakly decomposition (A,B,C) such that G(A) is a weakly component and G(V-A)=G(B)+G(C). 256 Theorem 2 provides an O(n+m) algorithm for building a weakly decomposition for an incomplete and connected graph. Algorithm for the weakly decomposition of a graph Input: A connected graph with at least two nonadjacent vertices, G=(V,E). Output: A partition V=(A,N,R) such that G(A) is connected, N=N(A), A~R=N(A). begin A := any set of vertices such that A ∪ N(A) ⊂ V N:=N(A) R:=V-A∪ N(A) while ( ∃ n∈N, ∃ r∈R such that nr∉E ) do A:=A∪ n N:=(N-{n}) ∪ (N(n) ∩ R) R:=R-(N(n) ∩ R) end One can observe that [A ]G is connected, NNA= G ( ) , R not is φ is an invariant of the algorithm. Corollary 1. If G is a connected graph and (A,N,R) a weakly decomposition with A weakly component then the following holds: ω (G)=max{ω ([N])+ω ([R]),ω ([A ∪ N])}. 4 Trivially perfect graphs 4.1 Basic properties In this subsection we establish the necessary and sufficient conditions for a graph to be an trivially perfect graph. Definition 3. (4) A graph G is trivially perfect if for each induced subgraph H of G, the number of maximal cliques of H is equal to the maximum size an independent set of H. Theorem 4. (4) A graph is trivially perfect if and only if it contains no vertex subset that induces P4 or C4. In [2, Brandstadt et al. establish: Trivially pergect graphs are exactly the chordal cographs. 257 A vertex ordering α = (v1 ,v2 ,…,vn ) of a graph G is a universal-in-a- component ordering (uco) if for 1 ≤ i ≤ n , the vertex vi is universal in the connected component of G({vi ,vi+1 ,…,vn }) that vi belongs to. A graph is trivially perfect if and only if it has a uco. A cograph G is a trivially perfect if and only if, in the cotree T of G, every 1-node has at most one child that is a 0-node. Theorem 5. Let G=(V,E) be connected with at least two nonadjacent vertices and (A,N,R) a weakly decomposition with A weakly component. G is {P4 ,C4 }-free graph if and only if: i) A∼N∼R ii) N is clique iii) G(A), G(R) are {P4 ,C4 }-free graphs. Proof. If G is {P4 ,C4 }-free graph then G(A), G(N), G(R) are {P4 ,C4 }-free graphs and A∼N∼R Because G is C4 -free results that G is triangulated, so N is clique. We suppose that i), ii) and iii) holds. Since A∼N∼R, N is clique and G(A), G(R) are {P4 ,C4 }-free graphs it follows that G is P4 -free. Since G(R) is {P4 ,C4 }-free it follows that G(R) is triangulated. If G-R is not triangulated graph then there exists Ck ⊆ G − R (k ≥ 4) . If k ≥ 5 then G-R it contains P4 . So C4 ⊆ G − R . Because N is clique and G(A) is C4 -free it follows that C4 ∩ A ≠ φ , C4 ∩ N ≠ φ . Because A∼N it follows that ⎟C4 ∩ A ⎟=⎟C4 ∩ N ⎟=1. 4.2 The recognition algorithm. Theorem 5 leads to the following recognition algorithm. Input: A connected graph with at least two nonadjacent vertices, G=(V,E). Output: An answer to the question: is G a trivially perfect graph begin L:={G}; {L a list of graphs While ( L ≠ φ ) Extrac an element H from L; Find a weakly decomposition (A,N,R) for H; 258 If ( A ≠ N or N ≠ R) then Return: G is not trivially perfect Else if N is not clique then Return: G is not trivially perfect Else introduce in L the connected components of G(A), G(R) incomplete Return: G is trivially perfect end From Theorem 5 results: Corollary 2. If G is a connected graph and (A,N,R) a weakly decomposition with A weakly component then G is trivially perfect if and only if A∼N∼R N is clique G(A), G(R) are {P4 ,C4 }-free graphs. Corollary 3. If G is a connected graph, trivially perfect and (A,N,R) a weakly decomposition with A weakly component then the following holds: α(G) =max{α(G(A)),α(G(GR))} ω(G) =⎟N⎟+max{ω(G(A)),ω(G(R))}. References [1] Berge, C., " Graphs", Nort-Holland, Amsterdam (1985) [2] A.Brandstadt, V.B.Le, and J.P.Spinrad, Graph classes: A survey, Philadelphia, SIAM, 1999. [3] C. Croitoru, E. Olaru, M. Talmaciu, "Confidentially connected graphs", The annals of the University "Dunarea de Jos" of Galati, Suppliment to Tome XVIII (XXII) 2000, Proceedings of the international conference "The risk in contemporany economy. [4] M.C.Golumbic, Trivially perfect graph, Discrete Math 24: 105-107, 1978. 259 [5] Talmaciu, M., " Decomposition Problems in the Graph Theory with Applications in Combinatorial Optimization", Ph.D.Thesis, University "Al.
Load more

Recommended publications
  • On Treewidth and Graph Minors
    On Treewidth and Graph Minors Daniel John Harvey Submitted in total fulfilment of the requirements of the degree of Doctor of Philosophy February 2014 Department of Mathematics and Statistics The University of Melbourne Produced on archival quality paper ii Abstract Both treewidth and the Hadwiger number are key graph parameters in structural and al- gorithmic graph theory, especially in the theory of graph minors. For example, treewidth demarcates the two major cases of the Robertson and Seymour proof of Wagner's Con- jecture. Also, the Hadwiger number is the key measure of the structural complexity of a graph. In this thesis, we shall investigate these parameters on some interesting classes of graphs. The treewidth of a graph defines, in some sense, how \tree-like" the graph is. Treewidth is a key parameter in the algorithmic field of fixed-parameter tractability. In particular, on classes of bounded treewidth, certain NP-Hard problems can be solved in polynomial time. In structural graph theory, treewidth is of key interest due to its part in the stronger form of Robertson and Seymour's Graph Minor Structure Theorem. A key fact is that the treewidth of a graph is tied to the size of its largest grid minor. In fact, treewidth is tied to a large number of other graph structural parameters, which this thesis thoroughly investigates. In doing so, some of the tying functions between these results are improved. This thesis also determines exactly the treewidth of the line graph of a complete graph. This is a critical example in a recent paper of Marx, and improves on a recent result by Grohe and Marx.
    [Show full text]
  • Lecture 10: April 20, 2005 Perfect Graphs
    Re-revised notes 4-22-2005 10pm CMSC 27400-1/37200-1 Combinatorics and Probability Spring 2005 Lecture 10: April 20, 2005 Instructor: L´aszl´oBabai Scribe: Raghav Kulkarni TA SCHEDULE: TA sessions are held in Ryerson-255, Monday, Tuesday and Thursday 5:30{6:30pm. INSTRUCTOR'S EMAIL: [email protected] TA's EMAIL: [email protected], [email protected] IMPORTANT: Take-home test Friday, April 29, due Monday, May 2, before class. Perfect Graphs k 1=k Shannon capacity of a graph G is: Θ(G) := limk (α(G )) : !1 Exercise 10.1 Show that α(G) χ(G): (G is the complement of G:) ≤ Exercise 10.2 Show that χ(G H) χ(G)χ(H): · ≤ Exercise 10.3 Show that Θ(G) χ(G): ≤ So, α(G) Θ(G) χ(G): ≤ ≤ Definition: G is perfect if for all induced sugraphs H of G, α(H) = χ(H); i. e., the chromatic number is equal to the clique number. Theorem 10.4 (Lov´asz) G is perfect iff G is perfect. (This was open under the name \weak perfect graph conjecture.") Corollary 10.5 If G is perfect then Θ(G) = α(G) = χ(G): Exercise 10.6 (a) Kn is perfect. (b) All bipartite graphs are perfect. Exercise 10.7 Prove: If G is bipartite then G is perfect. Do not use Lov´asz'sTheorem (Theorem 10.4). 1 Lecture 10: April 20, 2005 2 The smallest imperfect (not perfect) graph is C5 : α(C5) = 2; χ(C5) = 3: For k 2, C2k+1 imperfect.
    [Show full text]
  • 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]
  • Symmetry and Structure of Graphs
    Symmetry and Structure of graphs by Kovács Máté Submitted to Central European University Department of Department of Mathematics and its Applications In partial fulfillment of the requirements for the degree of Master of Science Supervisor: Dr. Hegedus˝ Pál CEU eTD Collection Budapest, Hungary 2014 I, the undersigned [Kovács Máté], candidate for the degree of Master of Science at the Central European University Department of Mathematics and its Applications, declare herewith that the present thesis is exclusively my own work, based on my research and only such external information as properly credited in notes and bibliography. I declare that no unidentified and illegitimate use was made of work of others, and no part the thesis infringes on any person’s or institution’s copyright. I also declare that no part the thesis has been submitted in this form to any other institution of higher education for an academic degree. Budapest, 9 May 2014 ————————————————— Signature CEU eTD Collection c by Kovács Máté, 2014 All Rights Reserved. ii Abstract The thesis surveys results on structure and symmetry of graphs. Structure and symmetry of graphs can be handled by graph homomorphisms and graph automorphisms - the two approaches are compatible. Two graphs are called homomorphically equivalent if there is a graph homomorphism between the two graphs back and forth. Being homomorphically equivalent is an equivalence relation, and every class has a vertex minimal element called the graph core. It turns out that transitive graphs have transitive cores. The possibility of a structural result regarding transitive graphs is investigated. We speculate that almost all transitive graphs are cores.
    [Show full text]
  • Induced Minors and Well-Quasi-Ordering Jaroslaw Blasiok, Marcin Kamiński, Jean-Florent Raymond, Théophile Trunck
    Induced minors and well-quasi-ordering Jaroslaw Blasiok, Marcin Kamiński, Jean-Florent Raymond, Théophile Trunck To cite this version: Jaroslaw Blasiok, Marcin Kamiński, Jean-Florent Raymond, Théophile Trunck. Induced minors and well-quasi-ordering. EuroComb: European Conference on Combinatorics, Graph Theory and Appli- cations, Aug 2015, Bergen, Norway. pp.197-201, 10.1016/j.endm.2015.06.029. lirmm-01349277 HAL Id: lirmm-01349277 https://hal-lirmm.ccsd.cnrs.fr/lirmm-01349277 Submitted on 27 Jul 2016 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. Available online at www.sciencedirect.com www.elsevier.com/locate/endm Induced minors and well-quasi-ordering Jaroslaw Blasiok 1,2 School of Engineering and Applied Sciences, Harvard University, United States Marcin Kami´nski 2 Institute of Computer Science, University of Warsaw, Poland Jean-Florent Raymond 2 Institute of Computer Science, University of Warsaw, Poland, and LIRMM, University of Montpellier, France Th´eophile Trunck 2 LIP, ENS´ de Lyon, France Abstract AgraphH is an induced minor of a graph G if it can be obtained from an induced subgraph of G by contracting edges. Otherwise, G is said to be H-induced minor- free.
    [Show full text]
  • Arxiv:1907.08517V3 [Math.CO] 12 Mar 2021
    RANDOM COGRAPHS: BROWNIAN GRAPHON LIMIT AND ASYMPTOTIC DEGREE DISTRIBUTION FRÉDÉRIQUE BASSINO, MATHILDE BOUVEL, VALENTIN FÉRAY, LUCAS GERIN, MICKAËL MAAZOUN, AND ADELINE PIERROT ABSTRACT. We consider uniform random cographs (either labeled or unlabeled) of large size. Our first main result is the convergence towards a Brownian limiting object in the space of graphons. We then show that the degree of a uniform random vertex in a uniform cograph is of order n, and converges after normalization to the Lebesgue measure on [0; 1]. We finally an- alyze the vertex connectivity (i.e. the minimal number of vertices whose removal disconnects the graph) of random connected cographs, and show that this statistics converges in distribution without renormalization. Unlike for the graphon limit and for the degree of a random vertex, the limiting distribution of the vertex connectivity is different in the labeled and unlabeled settings. Our proofs rely on the classical encoding of cographs via cotrees. We then use mainly combi- natorial arguments, including the symbolic method and singularity analysis. 1. INTRODUCTION 1.1. Motivation. Random graphs are arguably the most studied objects at the interface of com- binatorics and probability theory. One aspect of their study consists in analyzing a uniform random graph of large size n in a prescribed family, e.g. perfect graphs [MY19], planar graphs [Noy14], graphs embeddable in a surface of given genus [DKS17], graphs in subcritical classes [PSW16], hereditary classes [HJS18] or addable classes [MSW06, CP19]. The present paper focuses on uniform random cographs (both in the labeled and unlabeled settings). Cographs were introduced in the seventies by several authors independently, see e.g.
    [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]
  • A Comparison of Two Approaches for Polynomial Time Algorithms
    A comparison of two approaches for polynomial time algorithms computing basic graph parameters∗† Frank Gurski‡ March 26, 2018 Abstract In this paper we compare and illustrate the algorithmic use of graphs of bounded tree- width and graphs of bounded clique-width. For this purpose we give polynomial time algorithms for computing the four basic graph parameters independence number, clique number, chromatic number, and clique covering number on a given tree structure of graphs of bounded tree-width and graphs of bounded clique-width in polynomial time. We also present linear time algorithms for computing the latter four basic graph parameters on trees, i.e. graphs of tree-width 1, and on co-graphs, i.e. graphs of clique-width at most 2. Keywords:graph algorithms, graph parameters, clique-width, NLC-width, tree-width 1 Introduction A graph parameter is a mapping that associates every graph with a positive integer. Well known graph parameters are independence number, dominating number, and chromatic num- ber. In general the computation of such parameters for some given graph is NP-hard. In this work we give fixed-parameter tractable (fpt) algorithms for computing basic graph parameters restricted to graph classes of bounded tree-width and graph classes of bounded clique-width. The tree-width of graphs has been defined in 1976 by Halin [Hal76] and independently in arXiv:0806.4073v1 [cs.DS] 25 Jun 2008 1986 by Robertson and Seymour [RS86] by the existence of a tree decomposition. Intuitively, the tree-width of some graph G measures how far G differs from a tree. Two more powerful and more recent graph parameters are clique-width1 and NLC-width2 both defined in 1994, by Courcelle and Olariu [CO00] and by Wanke [Wan94], respectively.
    [Show full text]
  • 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.
    [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]
  • 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.
    [Show full text]
  • Partitioning a Graph Into Disjoint Cliques and a Triangle-Free Graph
    This is a repository copy of Partitioning a graph into disjoint cliques and a triangle-free graph. White Rose Research Online URL for this paper: http://eprints.whiterose.ac.uk/85292/ Version: Accepted Version Article: Abu-Khzam, FN, Feghali, C and Muller, H (2015) Partitioning a graph into disjoint cliques and a triangle-free graph. Discrete Applied Mathematics, 190-19. 1 - 12. ISSN 0166-218X https://doi.org/10.1016/j.dam.2015.03.015 © 2015, Elsevier. Licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International http://creativecommons.org/licenses/by-nc-nd/4.0/ Reuse Unless indicated otherwise, fulltext items are protected by copyright with all rights reserved. The copyright exception in section 29 of the Copyright, Designs and Patents Act 1988 allows the making of a single copy solely for the purpose of non-commercial research or private study within the limits of fair dealing. The publisher or other rights-holder may allow further reproduction and re-use of this version - refer to the White Rose Research Online record for this item. Where records identify the publisher as the copyright holder, users can verify any specific terms of use on the publisher’s website. Takedown If you consider content in White Rose Research Online to be in breach of UK law, please notify us by emailing [email protected] including the URL of the record and the reason for the withdrawal request. [email protected] https://eprints.whiterose.ac.uk/ Partitioning a Graph into Disjoint Cliques and a Triangle-free Graph Faisal N. Abu-Khzam, Carl Feghali, Haiko M¨uller 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.
    [Show full text]