Secure Integer Domination in Graphs

An Integer dominating function on a graph G is a function f : V (G) → W such that for every vertex v ∈ V (G),P v∈V (G)(N [v]) ≥ k. For any function f : V (G) → W and any pair of adjacent vertices with f (v) = 0 and u > 0, the function guv is defined by guv(l) = 1, guv(l) = f (u) − 1 and guv(l) = f (l) if l ∈ V − {u, v}. A secure integer dominating function on a graph G is defined as an integer dominating function g which satisfies the condition that for every vertex v with f (v) = 0, ∃ a neighbor u with f (u) > 0 is such that guv is an integer dominating function. The weight of f is w(f ) = P v∈V (G) f (v). The minimum weight among all the dominant secure integer functions in G is the number of secure integer domination in G. This paper is devoted to initiating the study of SIDF of a graph. In particular, we have studied about the general bounds for standard graphs like Complete graph, Star graph, wheel graph, path, cycle.