Domination Value in Graphs

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

Broadcast dimension of graphs

In this paper we initiate the study of broadcast dimension, a variant of metric dimension. Let G be a graph with vertex set V (G), and let d(u, w) denote the length of a u − w geodesic in G. For k ≥ 1, let dk (x, y) = min{d(x, y), k +1}. A function f: V (G) → Z+ ∪{0} is called a resolving broadcast of G if, for any distinct x, y ∈ V (G), there exists a vertex [Formula Presented]. The broadcast dimension, bdim(G), of G is the minimum of [Formula Presented] over all resolving broadcasts of G, where bcf (G) can be viewed as the total cost of the transmitters (of various strength) used in resolving the entire network described by the graph G. Note that bdim(G) reduces to adim(G) (the adjacency dimension of G, introduced by Jannesari and Omoomi in 2012) if the codomain of resolving broadcasts is restricted to {0, 1}. We determine its value for cycles, paths, and other families of graphs. We prove that bdim(G) = Ω(log n) for all graphs G of order n, and that the result is sharp up to a constant factor. We show that [Formula Presented] and can both be arbitrarily large, where dim(G) denotes the metric dimension of G. We also examine the effect of vertex deletion on both the adjacency dimension and the broadcast dimension of graphs