12 research outputs found
The total bondage number of grid graphs
The total domination number of a graph without isolated vertices is the
minimum number of vertices that dominate all vertices in . The total bondage
number of is the minimum number of edges whose removal enlarges
the total domination number. This paper considers grid graphs. An -grid
graph is defined as the cartesian product of two paths and
. This paper determines the exact values of and
, and establishes some upper bounds of .Comment: 15 pages with 4 figure
Common extremal graphs for three inequalities involving domination parameters
Let , and be the minimum degree, maximum degree and domination number of a graph , respectively. A partition of , all of whose classes are dominating sets in , is called a domatic partition of . The maximum number of classes of a domatic partition of is called the domatic number of , denoted . It is well known that , cite{ch}, and cite{berge}. In this paper, we investigate the graphs for which all the above inequalities become simultaneously equalities
Distances and Domination in Graphs
This book presents a compendium of the 10 articles published in the recent Special Issue “Distance and Domination in Graphs”. The works appearing herein deal with several topics on graph theory that relate to the metric and dominating properties of graphs. The topics of the gathered publications deal with some new open lines of investigations that cover not only graphs, but also digraphs. Different variations in dominating sets or resolving sets are appearing, and a review on some networks’ curvatures is also present
The bondage numbers and efficient dominations of vertex-transitive graphs
AbstractThe bondage number of a graph G is the minimum number of edges whose removal results in a graph with larger domination number. A dominating set D is called an efficient dominating set of G if |N-[v]∩D|=1 for every vertex v∈V(G). In this paper we establish a tight lower bound for the bondage number of a vertex-transitive graph. We also obtain upper bounds for regular graphs by investigating the relation between the bondage number and the efficient domination. As applications, we determine the bondage number for some circulant graphs and tori by characterizing the existence of efficient dominating sets in these graphs