Unique response strong Roman dominating functions of graphs

Abstract

Given a simple graph G = (V, E) with maximum degree ∆. Let (V 0 , V 1 , V 2 ) be an ordered partition of V , where2 + 1} is a unique response strong Roman dominating function (URStRDF) if it is both URStRF and StRDF. The unique response strong Roman domination number of G, denoted by u StR (G), is the minimum weight of a unique response strong Roman dominating function. In this paper we approach the problem of a Roman domination-type defensive strategy under multiple simultaneous attacks and begin with the study of several mathematical properties of this invariant. We obtain several bounds on such a parameter and give some realizability results for it. Moreover, for any tree T of order n ≥ 3 we prove the sharp bound u StR (T ) ≤ 8n 9 .