Abstract
A set D ⊆ V (G) is a dominating set of G if every vertex not in D is adjacent to at least one vertex in D. A dominating set of G of minimum cardinality is called a γ(G)-set. For each vertex v ∈ V (G), we define the domination value of v to be the number of γ(G)-sets to which v belongs. In this paper, we study some basic properties of the domination value function, thus initiating a local study of domination in graphs. Further, we characterize domination value for the Petersen graph, complete n-partite graphs, cycles, and paths.
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