Extremal Problems for Forests in Graphs and Hypergraphs
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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. -
Linear K-Arboricities on Trees
Discrete Applied Mathematics 103 (2000) 281–287 View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Note Linear k-arboricities on trees Gerard J. Changa; 1, Bor-Liang Chenb, Hung-Lin Fua, Kuo-Ching Huangc; ∗;2 aDepartment of Applied Mathematics, National Chiao Tung University, Hsinchu 300, Taiwan bDepartment of Business Administration, National Taichung Institue of Commerce, Taichung 404, Taiwan cDepartment of Applied Mathematics, Providence University, Shalu 433, Taichung, Taiwan Received 31 October 1997; revised 15 November 1999; accepted 22 November 1999 Abstract For a ÿxed positive integer k, the linear k-arboricity lak (G) of a graph G is the minimum number ‘ such that the edge set E(G) can be partitioned into ‘ disjoint sets and that each induces a subgraph whose components are paths of lengths at most k. This paper studies linear k-arboricity from an algorithmic point of view. In particular, we present a linear-time algo- rithm to determine whether a tree T has lak (T)6m. ? 2000 Elsevier Science B.V. All rights reserved. Keywords: Linear forest; Linear k-forest; Linear arboricity; Linear k-arboricity; Tree; Leaf; Penultimate vertex; Algorithm; NP-complete 1. Introduction All graphs in this paper are simple, i.e., ÿnite, undirected, loopless, and without multiple edges. A linear k-forest is a graph whose components are paths of length at most k.Alinear k-forest partition of G is a partition of the edge set E(G) into linear k-forests. The linear k-arboricity of G, denoted by lak (G), is the minimum size of a linear k-forest partition of G. -
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). -
Coloring Problems in Graph Theory Kacy Messerschmidt Iowa State University
Iowa State University Capstones, Theses and Graduate Theses and Dissertations Dissertations 2018 Coloring problems in graph theory Kacy Messerschmidt Iowa State University Follow this and additional works at: https://lib.dr.iastate.edu/etd Part of the Mathematics Commons Recommended Citation Messerschmidt, Kacy, "Coloring problems in graph theory" (2018). Graduate Theses and Dissertations. 16639. https://lib.dr.iastate.edu/etd/16639 This Dissertation is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Graduate Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. Coloring problems in graph theory by Kacy Messerschmidt A dissertation submitted to the graduate faculty in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Major: Mathematics Program of Study Committee: Bernard Lidick´y,Major Professor Steve Butler Ryan Martin James Rossmanith Michael Young The student author, whose presentation of the scholarship herein was approved by the program of study committee, is solely responsible for the content of this dissertation. The Graduate College will ensure this dissertation is globally accessible and will not permit alterations after a degree is conferred. Iowa State University Ames, Iowa 2018 Copyright c Kacy Messerschmidt, 2018. All rights reserved. TABLE OF CONTENTS LIST OF FIGURES iv ACKNOWLEDGEMENTS vi ABSTRACT vii 1. INTRODUCTION1 2. DEFINITIONS3 2.1 Basics . .3 2.2 Graph theory . .3 2.3 Graph coloring . .5 2.3.1 Packing coloring . .6 2.3.2 Improper coloring . -
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. -
Structural Parameterizations of Clique Coloring
Structural Parameterizations of Clique Coloring Lars Jaffke University of Bergen, Norway lars.jaff[email protected] Paloma T. Lima University of Bergen, Norway [email protected] Geevarghese Philip Chennai Mathematical Institute, India UMI ReLaX, Chennai, India [email protected] Abstract A clique coloring of a graph is an assignment of colors to its vertices such that no maximal clique is monochromatic. We initiate the study of structural parameterizations of the Clique Coloring problem which asks whether a given graph has a clique coloring with q colors. For fixed q ≥ 2, we give an O?(qtw)-time algorithm when the input graph is given together with one of its tree decompositions of width tw. We complement this result with a matching lower bound under the Strong Exponential Time Hypothesis. We furthermore show that (when the number of colors is unbounded) Clique Coloring is XP parameterized by clique-width. 2012 ACM Subject Classification Mathematics of computing → Graph coloring Keywords and phrases clique coloring, treewidth, clique-width, structural parameterization, Strong Exponential Time Hypothesis Digital Object Identifier 10.4230/LIPIcs.MFCS.2020.49 Related Version A full version of this paper is available at https://arxiv.org/abs/2005.04733. Funding Lars Jaffke: Supported by the Trond Mohn Foundation (TMS). Acknowledgements The work was partially done while L. J. and P. T. L. were visiting Chennai Mathematical Institute. 1 Introduction Vertex coloring problems are central in algorithmic graph theory, and appear in many variants. One of these is Clique Coloring, which given a graph G and an integer k asks whether G has a clique coloring with k colors, i.e. -
Choosability on H-Free Graphs?
Choosability on H-Free Graphs? Petr A. Golovach1, Pinar Heggernes1, Pim van 't Hof1, and Dani¨el Paulusma2 1 Department of Informatics, University of Bergen, Norway fpetr.golovach,pinar.heggernes,[email protected] 2 School of Engineering and Computing Sciences, Durham University, UK [email protected] Abstract. A graph is H-free if it has no induced subgraph isomorphic to H. We determine the computational complexity of the Choosability problem restricted to H-free graphs for every graph H that does not belong to fK1;3;P1 + P2;P1 + P3;P4g. We also show that if H is a linear forest, then the problem is fixed-parameter tractable when parameterized by k. Keywords. Choosability, H-free graphs, linear forest. 1 Introduction Graph coloring is without doubt one of the most fundamental concepts in both structural and algorithmic graph theory. The well-known Coloring problem, which takes as input a graph G and an integer k, asks whether G admits a k- coloring, i.e., a mapping c : V (G) ! f1; 2; : : : ; kg such that c(u) 6= c(v) whenever uv 2 E(G). The corresponding problem where k is fixed, i.e., not part of the input, is denoted by k-Coloring. Since graph coloring naturally arises in a vast number of theoretical and practical applications, it is not surprising that many variants and generalizations of the Coloring problem have been studied over the years. There are some very good surveys [1,28] and a book [22] on the subject. One well-known generalization of graph coloring, called List Coloring, was introduced by Vizing [29] and Erd}os,Rubin and Taylor [7]. -
On the Threshold-Width of Graphs Maw-Shang Chang 1 Ling-Ju Hung 1 Ton Kloks Sheng-Lung Peng 2
Journal of Graph Algorithms and Applications http://jgaa.info/ vol. 15, no. 2, pp. 253–268 (2011) On the Threshold-Width of Graphs Maw-Shang Chang 1 Ling-Ju Hung 1 Ton Kloks Sheng-Lung Peng 2 1Department of Computer Science and Information Engineering National Chung Cheng University Min-Hsiung, Chia-Yi 621, Taiwan 2Department of Computer Science and Information Engineering National Dong Hwa University Shoufeng, Hualien 97401, Taiwan Abstract For a graph class G, a graph G has G-width k if there are k independent sets N1,..., Nk in G such that G can be embedded into a graph H ∈G such that for every edge e in H which is not an edge in G, there exists an i such that both endpoints of e are in Ni. For the class TH of threshold graphs we show that TH-width is NP-complete and we present fixed-parameter algorithms. We also show that for each k, graphs of TH-width at most k are characterized by a finite collection of forbidden induced subgraphs. Submitted: Reviewed: Revised: Accepted: September 2010 January 2011 March 2011 April 2011 Final: Published: May 2011 July 2011 Article type: Communicated by: Regular paper D. Wagner This research was supported by the National Science Council of Taiwan, under grants NSC 98– 2218–E–194–004 and NSC 98–2811–E–194–006. Ton Kloks is supported by the National Science Council of Taiwan, under grants the National Science Council of Taiwan, under grants NSC 99–2218–E–007–016 and NSC 99–2811–E–007–044. -
Complexity Results for Rainbow Matchings
Complexity Results for Rainbow Matchings Van Bang Le1 and Florian Pfender2 1 Universität Rostock, Institut für Informatik 18051 Rostock, Germany [email protected] 2 University of Colorado at Denver, Department of Mathematics & Statistics Denver, CO 80202, USA [email protected] Abstract A rainbow matching in an edge-colored graph is a matching whose edges have distinct colors. We address the complexity issue of the following problem, max rainbow matching: Given an edge-colored graph G, how large is the largest rainbow matching in G? We present several sharp contrasts in the complexity of this problem. We show, among others, that max rainbow matching can be approximated by a polynomial algorithm with approxima- tion ratio 2/3 − ε. max rainbow matching is APX-complete, even when restricted to properly edge-colored linear forests without a 5-vertex path, and is solvable in polynomial time for edge-colored forests without a 4-vertex path. max rainbow matching is APX-complete, even when restricted to properly edge-colored trees without an 8-vertex path, and is solvable in polynomial time for edge-colored trees without a 7-vertex path. max rainbow matching is APX-complete, even when restricted to properly edge-colored paths. These results provide a dichotomy theorem for the complexity of the problem on forests and trees in terms of forbidding paths. The latter is somewhat surprising, since, to the best of our knowledge, no (unweighted) graph problem prior to our result is known to be NP-hard for simple paths. We also address the parameterized complexity of the problem. -
The Classification of B-Perfect Graphs
The classification of B-perfect graphs Stephan Dominique Andres Department of Mathematics and Computer Science, FernUniversit¨at in Hagen, Germany Key words: game chromatic number, game-perfectness, trivially perfect, graph colouring game 1 Introduction Consider the following game, played on an (initially uncolored) graph G = (V, E) with a color set C. The players, Alice and Bob, move alternately. A move consists in coloring a vertex v ∈ V with a color c ∈ C in such a way that adjacent vertices receive distinct colors. If this is not possible any more, the game ends. Alice wins if every vertex is colored in the end, otherwise Bob wins. This type of game was introduced by Bodlaender [2]. He considers a variant, which we will call game g, in which Alice must move first and passing is not allowed. In order to obtain upper and lower bounds for a parameter associated with game g, two other variants are useful. In the game B Bob may move first. He may also miss one or several turns, but Alice must always move. In the other variant, game A, Alice may move first and miss one or several turns, but Bob must move. So in game B Bob has some advantages, whereas in game A Alice has some advantages with respect to Bodlaender’s game. For any variant G∈{B,g,A}, the smallest cardinality of a color set C, so that Alice has a winning strategy for the game G is called G-game chromatic number χG(G) of G. Email address: [email protected] (Stephan Dominique Andres).