Changing and Unchanging 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), . For any function f : V (G) → W and any pair of adjacent vertices with f(v) = 0 and u > 0, the function  is defined by  (l) = 1,  (l) = f(u) − 1 and  (l) = f(l) if . A secure integer dominating function on a graph G is defined as an integer dominating function  which satisfies that for every vertex v with f(v) = 0, a neighbour u with f(u) > 0 such that is an integer dominating function. The weight of f is w(f) = . Minimum weight among all the secure integer dominating function on G is secure integer domination number on G. This paper is devoted to initiating the study of SIDF of a graph. In particular, we have studied the changing and unchanging behavior of the graphs. An Integer dominating function on a graph G is a function f : V (G) → W such that for every vertex  v ∈ V (G), . For any function f : V (G) → W and any pair of adjacent vertices with f(v) = 0 and u > 0, the function  is defined by  (l) = 1,  (l) = f(u) − 1 and  (l) = f(l) if . A secure integer dominating function on a graph G is defined as an integer dominating function  which satisfies that for every vertex v with f(v) = 0, a neighbour u with f(u) > 0 such that is an integer dominating function. The weight of f is w(f) = . Minimum weight among all the secure integer dominating function on G is secure integer domination number on G. This paper is devoted to initiating the study of SIDF of a graph. In particular, we have studied the changing and unchanging behavior of the graphs.   Objectives: We propose a novel generalization of domination, which incorporates additional security and broader applicability. This refined framework offers new possibilities for research and practical implementation. Objectives: We propose a novel generalization of domination, which incorporates additional security and broader applicability. This refined framework offers new possibilities for research and practical implementation.