From the Strong Differential to Italian Domination in Graphs

Identifier:  imarina:9228569

Handle: https://hdl.handle.net/20.500.11797/imarina9228569

Authors:  Cabrera Martinez, A; Rodriguez-Velazquez, J A

Abstract:
Given a graph G and a subset of vertices D subset of V (G), the external neighbourhood of D is defined as N-e(D) = {u is an element of V (G)\D : N(u) boolean AND D not equal O}, where N(u) denotes the open neighbourhood of u. Now, given a subset D subset of V (G) and a vertex v. D, the external private neighbourhood of v with respect to D is defined to be epn(v, D) = {u is an element of V(G)\D : N(u)boolean AND D = {v}}. The strong differential of a set D subset of V (G) is defined as partial derivative(s)(D) = vertical bar N-e(D)vertical bar - vertical bar D-w vertical bar, where D-w = {v is an element of D : epn(v, D) v not equal O}. In this paper, we focus on the study of the strong differential of a graph, which is defined aspartial derivative(s)(G) = max{partial derivative(s)(D) : D subset of V (G)}.Among other results, we obtain general bounds on partial derivative(s)(G) and we prove a Gallai-type theorem, which states that partial derivative(s)(G) + gamma(I)(G) = n(G), where gamma(I)G) denotes the Italian domination number of G. Therefore, we can see the theory of strong differential in graphs as a new approach to the theory of Italian domination. One of the advantages of this approach is that it allows us to study the Italian domination number without the use of functions. As we can expect, we derive new results on the Italian domination number of a graph.