Affiliations: [a] Department of Mathematics, Karabuk University, 78050 Karabuk, Turkey. [email protected] | [b] Department of Mathematics, Ege University, 35100 İzmir, Turkey. [email protected]
Correspondence:
[*]
Address for correspondence: Department of Mathematics, Faculty of Science, Karabuk University, 78050 Karabuk,Turkey.
Abstract: Let G = (V (G), E(G)) be a simple undirected graph. The domination and average lower domination numbers are vulnerability parameters of a graph. We have investigated a refinement that involves the residual domination and average lower residual domination numbers of these parameters. The lower residual domination number, denoted by γvkR(G)
, is the minimum cardinality of dominating set in G that received from the graph G where the vertex vk and all links of the vertex vk are deleted. The residual domination number of graphs G is defined as γR(G)=minvk∈V(G){γvkR(G)}
. The average lower residual domination number of G is defined by γavR(G)=1|V(G)|∑vk∈V(G)γVkR(G)
. In this paper, we define the residual domination and the average lower residual domination numbers of a graph and we present the exact values, upper and lower bounds for some graph families.
Keywords: Network Design and Communication, Graph vulnerability, Domination number, Residual domination number, Average lower residual domination number
