SECURE INTEGER DOMINATION FOR FORBIDDEN GRAPHS

An Integer dominating function on a graph G is a function f : V(G) → W such that for every vertex 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 (SIDF) 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  . 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 secure integer domination for forbidden graphs.