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Characterization of graphs dominated by induced paths

G. Bacsó, Zs. Tuza, M. Voigt
2007 Discrete Mathematics  
We give a characterization, in terms of forbidden induced subgraphs, of those graphs in which every connected induced subgraph has a dominating induced path on at most k vertices (k 3).  ...  We show, in particular, that k = 4 means precisely the class of domination-reducible graphs, whose original definition applied four types of structural reduction.  ...  The class of domination-reducible graphs is precisely the class of hereditarily D 4 -dominated graphs. Corollary 3.  ... 
doi:10.1016/j.disc.2005.11.035 fatcat:lflphzl7ovbmdfwdimgthoapke

Characterization The Deletable Set Of Vertices In The ( P - 3 )- Regular Graphs

Akram B. Attar
2011 Journal of Mathematics and Computer Science  
In this paper we characterized the ( p -3 )-regular graphs which have a 3−deletable and a 4−deletable set of vertices. (2000) : 05C35,05C69, 05C60. Mathematics Subject Classification  ...  Characterization the deletable set of vertices In this section we characterized the − 3 − regular graphs which contain a 3 − and a 4 − set of vertices.  ...  Introduction The roots of our study in deletable set of vertices are in the problem of reducibility of graphs.  ... 
doi:10.22436/jmcs.03.02.07 fatcat:2ce675pxgfg4xpxwvefptoftdq

Characterizations of minimal dominating sets and the well-dominated property in lexicographic product graphs [article]

Didem Gözüpek, Ademir Hujdurović, Martin Milanič
2017 arXiv   pre-print
Based on this notion, we characterize the family of minimal dominating sets in a lexicographic product of two graphs and derive a characterization of the well-dominated lexicographic product graphs.  ...  As a side result motivated by this study, we give a polynomially testable characterization of well-dominated graphs with domination number two, and show, more generally, that well-dominated graphs can  ...  The main application of this notion was a characterization of the minimal dominating sets in nontrivial lexicographic product graphs, which led to a complete characterization of nontrivial lexicographic  ... 
arXiv:1701.05886v3 fatcat:hgkaqvgcxfhnzhg7ig34vh5yke

The Inapproximability of Non NP-hard Optimization Problems [chapter]

Liming Cai, David Juedes, Iyad Kanj
1998 Lecture Notes in Computer Science  
Dominating Set: Given a tournament graph G, nd a minimum dominating set for the graph.  ...  Log k Dominating Set: (k 1) Given a directed graph G = (V; E) in which V = fv 1 ; : : :; v n g and subset fv 1 ; ; v log k n g of V forms a dominating set for G, nd a minimum dominating set for G.  ...  It reduces any graph G to a hypergraph H such that any vertex cover of size m for H can be used to construct in polynomial time a dominating set of size m for graph G.  ... 
doi:10.1007/3-540-49381-6_46 fatcat:zstrtvcwnzebbb5qmjl23qkxkq

Characterization of block graphs with equal 2-domination number and domination number plus one

Adriana Hansberg Pastor, Lutz Volkmann
2007 Discussiones Mathematicae Graph Theory  
Let G be a simple graph, and let p be a positive integer. A subset is the minimum cardinality among the p-dominating sets of G.  ...  If G is a nontrivial connected block graph, then we show that γ 2 (G) ≥ γ(G) + 1, and we characterize all connected block graphs with γ 2 (G) = γ(G) + 1.  ...  In this case we can reduce G to a graph G that consists of G 0 , the blocks B 1 , B 2 , B 3 and with either a leaf to the vertex v or to the vertex x.  ... 
doi:10.7151/dmgt.1347 fatcat:4witw6cinvbzxesdi75bbastky

The inapproximability of non-NP-hard optimization problems

Liming Cai, David Juedes, Iyad Kanj
2002 Theoretical Computer Science  
A direct connection is established between non-NP-hard problems and a PCP characterization of NP.  ...  A number of nontrivial approximation-preserving reductions are also presented, making it possible to extend inapproximability results to more natural non-NP-hard problems such as TOURNAMENT DOMINATING  ...  We reduce the PCP characterization of a language L ∈ NP in Corollary 3:6 to MAX SPARSE SAT.  ... 
doi:10.1016/s0304-3975(01)00343-7 fatcat:mkrjr5hw3fhh3f3lqaznumhp24

Page 1555 of Mathematical Reviews Vol. , Issue 2002C [page]

2002 Mathematical Reviews  
In this article, we provide a constructive characterization of those trees with equal independent domination and restrained domination numbers.  ...  A constructive characterization of those trees with equal independent domination and weak domination numbers is also obtained.” 2002¢:05053 05C05 05C20 Ziolo, Irmina A.  ... 

Page 2348 of Mathematical Reviews Vol. , Issue 2003d [page]

2003 Mathematical Reviews  
Given a set A CN, we characterize cycles C, with sets S C V(C,) that are simultaneously a-step dominating for precisely those aA.  ...  The first type is formed by reducing the num- ber of vertices which must be dominated by a pair of vertices and the second by increasing the number of steps allowable for dom- ination.  ... 

Efficient domination in circulant graphs

K. Reji Kumar, Gary MacGillivray
2013 Discrete Mathematics  
It is proved that if a circulant graph of large degree has an efficient dominating set, then either its elements are equally spaced, or the graph is the wreath product of a smaller circulant graph with  ...  an efficient dominating set and a complete graph.  ...  Acknowledgment The research of the second author was supported by NSERC.  ... 
doi:10.1016/j.disc.2012.12.003 fatcat:sem64ndpibf2hnx3w6faeyzz3q

Preface

Sandi Klavžar, Xueliang Li, Sanming Zhou
2016 Bulletin of the Malaysian Mathematical Sciences Society  
To reflect recent developments in graph theory in a timely manner and also to reduce the backlog of the Bulletin, we decided to edit this special issue on graph theory.  ...  The first contains papers of algebraic flavor, starting with a paper addressing graphs of ideals of a commutative ring.  ...  One of the deepest topics studied in this area is the Gromov hyperbolicity, and the first paper in this part gives a characterization of the hyperbolicity of periodic graphs.  ... 
doi:10.1007/s40840-015-0302-2 fatcat:m7q6g3z7x5gg7kqgaoqk4wn4ve

Majorizations for the dominant eigenvector of a nonnegative matrix

Roy Sandip, Ali Saberi, Yan Wan
2008 2008 American Control Conference  
Motivated by network controller design applications, we develop several majorization results for the dominant eigenvector of an irreducible nonnegative matrix.  ...  However, graph-theoretic characterization of eigenvalues and especially eigenvectors of asymmetric matrices is very limited.  ...  graph), and 2) the scaling of a row in the matrix (which corresponds to scaling the "influence" of a node in the graph, e.g. by changing a controller gain): The result for row-scaling of positive matrices  ... 
doi:10.1109/acc.2008.4586780 dblp:conf/amcc/RoySW08a fatcat:uc5t4n4qofccncp5fgxytsermi

Page 4155 of Mathematical Reviews Vol. , Issue 92h [page]

1992 Mathematical Reviews  
Summary: “A dominating set S of a graph G is perfect if each vertex of G is dominated by exactly one vertex in S.  ...  We link intersection graphs with cographs by introduc- ing ‘component-reducible multigraphs’ (multigraph analogues of cographs) and relate them to what we call ‘domination-reducible graphs’ (also known  ... 

Page 5188 of Mathematical Reviews Vol. , Issue 96i [page]

1996 Mathematical Reviews  
A set of six graph reduction operations is defined, and it is shown that a graph G € Y iff G can be reduced to an edgeless graph by a finite sequence of these operations.  ...  (BE-BEL; Minsk) An induced subgraph characterization of domination perfect graphs. (English summary) : J. Graph Theory 20 (1995), no. 3, 375-395.  ... 

A witness version of the Cops and Robber game

Nancy E. Clarke
2009 Discrete Mathematics  
In a special case, this recognition theorem provides a characterization.  ...  In the case when sightings occur at regular intervals, we present a recognition theorem for graphs on which a single cop suffices to guarantee a win.  ...  Finally, we note that the proof of Theorem 9 establishes a characterization of triangle-free (k, 1)-win graphs, namely: A triangle-free graph G is (k, 1)-win if and only if G can be reduced to a singleton  ... 
doi:10.1016/j.disc.2008.09.032 fatcat:ejvwduuto5er5kmki6eyo4kkgq

Minimal Dominating Set Enumeration [chapter]

Mamadou Moustapha Kanté, Lhouari Nourine
2014 Encyclopedia of Algorithms  
Notice that if a class of graphs C has a polynomially bounded number of minimal separators, then the enumeration of minimal connected dominating sets can be reduced to Dom-Enum [13] . 5.  ...  For instance it is proved in [11; 13] that every minimal dominating set D of a split graph G can be characterized by D ∩ C(G) where C(G) is the clique of G.  ... 
doi:10.1007/978-3-642-27848-8_721-1 fatcat:52gg7tmopbaujf4b3femw4odra
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