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On the Super Domination Number of Graphs
On the super domination number of graphs Douglas J. Klein1, Juan A. Rodr´ıguez-Vel´azquez1,2, Eunjeong Yi1 1Texas A&M University at Galveston Foundational Sciences, Galveston, TX 77553, United States 2Universitat Rovira i Virgili, Departament d’Enginyeria Inform`atica i Matem`atiques Av. Pa¨ısos Catalans 26, 43007 Tarragona, Spain Email addresses: [email protected], [email protected],[email protected] April 24, 2018 Abstract The open neighbourhood of a vertex v of a graph G is the set N(v) consisting of all vertices adjacent to v in G. For D ⊆ V (G), we define D = V (G) \ D. A set D ⊆ V (G) is called a super dominating set of G if for every vertex u ∈ D, there exists v ∈ D such that N(v) ∩ D = {u}. The super domination number of G is the minimum cardinality among all super dominating sets in G. In this article, we obtain closed formulas and tight bounds for the super domination number of G in terms of several invariants of G. Furthermore, the particular cases of corona product graphs and Cartesian product graphs are considered. Keywords: Super domination number; Domination number; Cartesian product; Corona product. AMS Subject Classification numbers: 05C69; 05C70 ; 05C76 1 Introduction arXiv:1705.00928v2 [math.CO] 23 Apr 2018 The open neighbourhood of a vertex v of a graph G is the set N(v) consisting of all vertices adjacent to v in G. For D ⊆ V (G), we define D = V (G) \ D. A set D ⊆ V (G) is dominating in G if every vertex in D has at least one neighbour in D, i.e., N(u) ∩ D =6 ∅ for every u ∈ D. -
Merging Peg Solitaire in Graphs
Marquette University e-Publications@Marquette Mathematics, Statistics and Computer Science Mathematics, Statistics and Computer Science, Faculty Research and Publications Department of (-2019) 7-17-2017 Merging Peg Solitaire in Graphs John Engbers Marquette University, [email protected] Ryan Weber Marquette University Follow this and additional works at: https://epublications.marquette.edu/mscs_fac Part of the Computer Sciences Commons, Mathematics Commons, and the Statistics and Probability Commons Recommended Citation Engbers, John and Weber, Ryan, "Merging Peg Solitaire in Graphs" (2017). Mathematics, Statistics and Computer Science Faculty Research and Publications. 629. https://epublications.marquette.edu/mscs_fac/629 INVOLVE 11:1 (2018) :-msp dx.doi.org/10.2140/involve.2018.11.53 Merging peg solitaire on graphs John Engbers and Ryan Weber (Communicated by Anant Godbole) Peg solitaire has recently been generalized to graphs. Here, pegs start on all but one of the vertices in a graph. A move takes pegs on adjacent vertices x and y, with y also adjacent to a hole on vertex z, and jumps the peg on x over the peg on y to z, removing the peg on y. The goal of the game is to reduce the number of pegs to one. We introduce the game merging peg solitaire on graphs, where a move takes pegs on vertices x and z (with a hole on y) and merges them to a single peg on y. When can a confguration on a graph, consisting of pegs on all vertices but one, be reduced to a confguration with only a single peg? We give results for a number of graph classes, including stars, paths, cycles, complete bipartite graphs, and some caterpillars. -
Decomposition of Wheel Graphs Into Stars, Cycles and Paths
Malaya Journal of Matematik, Vol. 9, No. 1, 456-460, 2021 https://doi.org/10.26637/MJM0901/0076 Decomposition of wheel graphs into stars, cycles and paths M. Subbulakshmi1 and I. Valliammal2* Abstract Let G = (V;E) be a finite graph with n vertices. The Wheel graph Wn is a graph with vertex set fv0;v1;v2;:::;vng and edge-set consisting of all edges of the form vivi+1 and v0vi where 1 ≤ i ≤ n, the subscripts being reduced modulo n. Wheel graph of (n + 1) vertices denoted by Wn. Decomposition of Wheel graph denoted by D(Wn). A star with 3 edges is called a claw S3. In this paper, we show that any Wheel graph can be decomposed into following ways. 8 (n − 2d)S ; d = 1;2;3;::: if n ≡ 0 (mod 6) > 3 > >[(n − 2d) − 1]S3 and P3; d = 1;2;3::: if n ≡ 1 (mod 6) > <[(n − 2d) − 1]S3 and P2; d = 1;2;3;::: if n ≡ 2 (mod 6) D(Wn) = . (n − 2d)S and C ; d = 1;2;3;::: if n ≡ 3 (mod 6) > 3 3 > >(n − 2d)S3 and P3; d = 1;2;3::: if n ≡ 4 (mod 6) > :(n − 2d)S3 and P2; d = 1;2;3;::: if n ≡ 5 (mod 6) Keywords Wheel Graph, Decomposition, claw. AMS Subject Classification 05C70. 1Department of Mathematics, G.V.N. College, Kovilpatti, Thoothukudi-628502, Tamil Nadu, India. 2Department of Mathematics, Manonmaniam Sundaranar University, Tirunelveli-627012, Tamil Nadu, India. *Corresponding author: [email protected]; [email protected] Article History: Received 01 November 2020; Accepted 30 January 2021 c 2021 MJM. -
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). -
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. -
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. -
Coalition Graphs of Paths, Cycles, and Trees
Discussiones Mathematicae Graph Theory xx (xxxx) 1–16 https://doi.org/10.7151/dmgt.2416 COALITION GRAPHS OF PATHS, CYCLES, AND TREES Teresa W. Haynes Department of Mathematics and Statistics East Tennessee State University Johnson City, TN 37604 e-mail: [email protected] Jason T. Hedetniemi Department of Mathematics Wilkes Honors College Florida Atlantic University Jupiter, FL 33458 e-mail: [email protected] Stephen T. Hedetniemi School of Computing Clemson University Clemson, SC 29634 e-mail: [email protected] Alice A. McRae and Raghuveer Mohan Computer Science Department Appalachian State University Boone, NC 28608 e-mail: [email protected] [email protected] Abstract A coalition in a graph G =(V,E) consists of two disjoint sets of vertices V1 and V2, neither of which is a dominating set of G but whose union V1 ∪V2 is a dominating set of G. A coalition partition in a graph G of order n = |V | 2 Haynes, Hedetniemi, Hedetniemi, McRae and Mohan is a vertex partition π = {V1,V2,...,Vk} of V such that every set Vi either is a dominating set consisting of a single vertex of degree n − 1, or is not a dominating set but forms a coalition with another set Vj which is not a dominating set. Associated with every coalition partition π of a graph G is a graph called the coalition graph of G with respect to π, denoted CG(G, π), the vertices of which correspond one-to-one with the sets V1,V2,...,Vk of π and two vertices are adjacent in CG(G, π) if and only if their corresponding sets in π form a coalition. -
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. -
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). -
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. -
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. -
Arxiv:1906.05510V2 [Math.AC]
BINOMIAL EDGE IDEALS OF COGRAPHS THOMAS KAHLE AND JONAS KRUSEMANN¨ Abstract. We determine the Castelnuovo–Mumford regularity of binomial edge ideals of complement reducible graphs (cographs). For cographs with n vertices the maximum regularity grows as 2n/3. We also bound the regularity by graph theoretic invariants and construct a family of counterexamples to a conjecture of Hibi and Matsuda. 1. Introduction Let G = ([n], E) be a simple undirected graph on the vertex set [n]= {1,...,n}. x1 ··· xn x1 ··· xn Let X = ( y1 ··· yn ) be a generic 2 × n matrix and S = k[ y1 ··· yn ] the polynomial ring whose indeterminates are the entries of X and with coefficients in a field k. The binomial edge ideal of G is JG = hxiyj −yixj : {i, j} ∈ Ei⊆ S, the ideal of 2×2 mi- nors indexed by the edges of the graph. Since their inception in [5, 15], connecting combinatorial properties of G with algebraic properties of JG or S/ JG has been a popular activity. Particular attention has been paid to the minimal free resolution of S/ JG as a standard N-graded S-module [3, 11]. The data of a minimal free res- olution is encoded in its graded Betti numbers βi,j (S/ JG) = dimk Tori(S/ JG, k)j . An interesting invariant is the highest degree appearing in the resolution, the Castelnuovo–Mumford regularity reg(S/ JG) = max{j − i : βij (S/ JG) 6= 0}. It is a complexity measure as low regularity implies favorable properties like vanish- ing of local cohomology. Binomial edge ideals have square-free initial ideals by arXiv:1906.05510v2 [math.AC] 10 Mar 2021 [5, Theorem 2.1] and, using [1], this implies that the extremal Betti numbers and regularity can also be derived from those initial ideals.