Linear Kernel for Planar Connected Dominating Set
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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). -
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. -
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. -
List of NP-Complete Problems from Wikipedia, the Free Encyclopedia
List of NP -complete problems - Wikipedia, the free encyclopedia Page 1 of 17 List of NP-complete problems From Wikipedia, the free encyclopedia Here are some of the more commonly known problems that are NP -complete when expressed as decision problems. This list is in no way comprehensive (there are more than 3000 known NP-complete problems). Most of the problems in this list are taken from Garey and Johnson's seminal book Computers and Intractability: A Guide to the Theory of NP-Completeness , and are here presented in the same order and organization. Contents ■ 1 Graph theory ■ 1.1 Covering and partitioning ■ 1.2 Subgraphs and supergraphs ■ 1.3 Vertex ordering ■ 1.4 Iso- and other morphisms ■ 1.5 Miscellaneous ■ 2 Network design ■ 2.1 Spanning trees ■ 2.2 Cuts and connectivity ■ 2.3 Routing problems ■ 2.4 Flow problems ■ 2.5 Miscellaneous ■ 2.6 Graph Drawing ■ 3 Sets and partitions ■ 3.1 Covering, hitting, and splitting ■ 3.2 Weighted set problems ■ 3.3 Set partitions ■ 4 Storage and retrieval ■ 4.1 Data storage ■ 4.2 Compression and representation ■ 4.3 Database problems ■ 5 Sequencing and scheduling ■ 5.1 Sequencing on one processor ■ 5.2 Multiprocessor scheduling ■ 5.3 Shop scheduling ■ 5.4 Miscellaneous ■ 6 Mathematical programming ■ 7 Algebra and number theory ■ 7.1 Divisibility problems ■ 7.2 Solvability of equations ■ 7.3 Miscellaneous ■ 8 Games and puzzles ■ 9 Logic ■ 9.1 Propositional logic ■ 9.2 Miscellaneous ■ 10 Automata and language theory ■ 10.1 Automata theory http://en.wikipedia.org/wiki/List_of_NP-complete_problems 12/1/2011 List of NP -complete problems - Wikipedia, the free encyclopedia Page 2 of 17 ■ 10.2 Formal languages ■ 11 Computational geometry ■ 12 Program optimization ■ 12.1 Code generation ■ 12.2 Programs and schemes ■ 13 Miscellaneous ■ 14 See also ■ 15 Notes ■ 16 References Graph theory Covering and partitioning ■ Vertex cover [1][2] ■ Dominating set, a.k.a. -
Dominating Cliques in Graphs
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Discrete Mathematics 86 (1990) 101-116 101 North-Holland DOMINATING CLIQUES IN GRAPHS Margaret B. COZZENS Mathematics Department, Northeastern University, Boston, MA 02115, USA Laura L. KELLEHER Massachusetts Maritime Academy, USA Received 2 December 1988 A set of vertices is a dominating set in a graph if every vertex not in the dominating set is adjacent to one or more vertices in the dominating set. A dominating clique is a dominating set that induces a complete subgraph. Forbidden subgraph conditions sufficient to imply the existence of a dominating clique are given. For certain classes of graphs, a polynomial algorithm is given for finding a dominating clique. A forbidden subgraph characterization is given for a class of graphs that have a connected dominating set of size three. Introduction A set of vertices in a simple, undirected graph is a dominating set if every vertex in the graph which is not in the dominating set is adjacent to one or more vertices in the dominating set. The domination number of a graph G is the minimum number of vertices in a dominating set. Several types of dominating sets have been investigated, including independent dominating sets, total dominating sets, and connected dominating sets, each of which has a corresponding domination number. Dominating sets have applications in a variety of fields, including communication theory and political science. For more background on dominating sets see [3, 5, 151 and other articles in this issue. For arbitrary graphs, the problem of finding the size of a minimum dominating set in the graph is an NP-complete problem [9]. -
The Bidimensionality Theory and Its Algorithmic
The Bidimensionality Theory and Its Algorithmic Applications by MohammadTaghi Hajiaghayi B.S., Sharif University of Technology, 2000 M.S., University of Waterloo, 2001 Submitted to the Department of Mathematics in partial ful¯llment of the requirements for the degree of DOCTOR OF PHILOSOPHY at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 2005 °c MohammadTaghi Hajiaghayi, 2005. All rights reserved. The author hereby grants to MIT permission to reproduce and distribute publicly paper and electronic copies of this thesis document in whole or in part. Author.............................................................. Department of Mathematics April 29, 2005 Certi¯ed by. Erik D. Demaine Associate Professor of Electrical Engineering and Computer Science Thesis Supervisor Accepted by . Rodolfo Ruben Rosales Chairman, Applied Mathematics Committee Accepted by . Pavel I. Etingof Chairman, Department Committee on Graduate Students 2 The Bidimensionality Theory and Its Algorithmic Applications by MohammadTaghi Hajiaghayi Submitted to the Department of Mathematics on April 29, 2005, in partial ful¯llment of the requirements for the degree of DOCTOR OF PHILOSOPHY Abstract Our newly developing theory of bidimensional graph problems provides general techniques for designing e±cient ¯xed-parameter algorithms and approximation algorithms for NP- hard graph problems in broad classes of graphs. This theory applies to graph problems that are bidimensional in the sense that (1) the solution value for the k £ k grid graph (and similar graphs) grows with k, typically as (k2), and (2) the solution value goes down when contracting edges and optionally when deleting edges. Examples of such problems include feedback vertex set, vertex cover, minimum maximal matching, face cover, a series of vertex- removal parameters, dominating set, edge dominating set, r-dominating set, connected dominating set, connected edge dominating set, connected r-dominating set, and unweighted TSP tour (a walk in the graph visiting all vertices). -
Algorithms for Minimum M-Connected K-Tuple Dominating Set Problem$
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Theoretical Computer Science 381 (2007) 241–247 www.elsevier.com/locate/tcs Algorithms for minimum m-connected k-tuple dominating set problem$ Weiping Shanga,c,∗, Pengjun Wanb, Frances Yaoc, Xiaodong Hua a Institute of Applied Mathematics, Chinese Academy of Sciences, Beijing 10080, China b Department of Computer Science, Illinois Institute of Technology, Chicago, IL 60616, USA c Department of Computer Science, City University of Hong Kong, Hong Kong Received 24 January 2007; received in revised form 25 April 2007; accepted 29 April 2007 Communicated by D.-Z. Du Abstract In wireless sensor networks, a virtual backbone has been proposed as the routing infrastructure to alleviate the broadcasting storm problem and perform some other tasks such as area monitoring. Previous work in this area has mainly focused on how to set up a small virtual backbone for high efficiency, which is modelled as the minimum Connected Dominating Set (CDS) problem. In this paper we consider how to establish a small virtual backbone to balance efficiency and fault tolerance. This problem can be formalized as the minimum m-connected k-tuple dominating set problem, which is a general version of minimum CDS problem with m = 1 and k = 1. We propose three centralized algorithms with small approximation ratios for small m and improve the current best results for small k. c 2007 Elsevier B.V. All rights reserved. Keywords: Connected dominating set; Approximation algorithm; k-vertex connectivity; Wireless sensor networks 1. Introduction A Wireless Sensor Network (WSN) consists of wireless nodes (transceivers) without any underlying physical infrastructure. -
Exponential Time Algorithms: Structures, Measures, and Bounds Serge Gaspers
Exponential Time Algorithms: Structures, Measures, and Bounds Serge Gaspers February 2010 2 Preface Although I am the author of this book, I cannot take all the credit for this work. Publications This book is a revised and updated version of my PhD thesis. It is partly based on the following papers in conference proceedings or journals. [FGPR08] F. V. Fomin, S. Gaspers, A. V. Pyatkin, and I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms, Algorithmica 52(2) (2008), 293{ 307. A preliminary version appeared in the proceedings of IWPEC 2006 [FGP06]. [GKL08] S. Gaspers, D. Kratsch, and M. Liedloff, On independent sets and bicliques in graphs, Proceedings of WG 2008, Springer LNCS 5344, Berlin, 2008, pp. 171{182. [GS09] S. Gaspers and G. B. Sorkin, A universally fastest algorithm for Max 2-Sat, Max 2- CSP, and everything in between, Proceedings of SODA 2009, ACM and SIAM, 2009, pp. 606{615. [GKLT09] S. Gaspers, D. Kratsch, M. Liedloff, and I. Todinca, Exponential time algorithms for the minimum dominating set problem on some graph classes, ACM Transactions on Algorithms 6(1:9) (2009), 1{21. A preliminary version appeared in the proceedings of SWAT 2006 [GKL06]. [FGS07] F. V. Fomin, S. Gaspers, and S. Saurabh, Improved exact algorithms for counting 3- and 4-colorings, Proceedings of COCOON 2007, Springer LNCS 4598, Berlin, 2007, pp. 65{74. [FGSS09] F. V. Fomin, S. Gaspers, S. Saurabh, and A. A. Stepanov, On two techniques of combining branching and treewidth, Algorithmica 54(2) (2009), 181{207. A preliminary version received the Best Student Paper award at ISAAC 2006 [FGS06]. -
A Dominating-Set-Based Routing Scheme in Ad Hoc Wireless
A Dominating-Set-Based Routing Scheme in Ad Ho c Wireless 1 Networks Jie Wu and Hailan Li Department of Computer Science and Engineering Florida Atlantic University Bo ca Raton, FL 33431 1 This work was supp orted in part by NSF grants CCR 9900646 and ANI 0073736. Abstract Ecient routing among a set of mobile hosts also called no des is one of the most imp ortant functions in ad-ho c wireless networks. Routing based on a connected dominating set is a promising approach, where the searching space for a route is reduced to no des in the set. A set is dominating if all the no des in the system are either in the set or neighb ors of no des in the set. In this pap er, we prop ose a simple and ecient distributed algorithm for calculating connected dominating set in ad- ho c wireless networks, where connections of no des are determined by their geographical distances. We also prop ose an up date/recalculation algorithm for the connected dominating set when the top ology of the ad ho c wireless network changes dynamically. Our simulation results show that the prop osed approach outp erforms a classical algorithm in terms of nding a small connected dominating set and doing so quickly. Our approach can be p otentially used in designing ecient routing algorithms based on a connected dominating set. Keywords: ad hoc wireless networks, dominating sets, mobile computing, routing, simulation 1 Intro duction Recent advances in technology have provided p ortable computers with wireless interfaces that allow networked communication among mobile users.