Unique response strong Roman dominating functions of graphs

Abstract

Given a simple graph G=(V,E) with maximum degree Δ . Let ( V 0 , V 1 , V 2 ) be an ordered partition of V, where V i = { v ∈ V : f ( v )= i } for i = 0, 1 and V 2 = { v ∈ V : f ( v )≥2} . A function f : V → {0, 1, …, ⌈ Δ /2⌉+1} is a strong Roman dominating function (StRDF) on G, if every v ∈ V 0 has a neighbor w ∈ V 2 and f ( w )≥1 + ⌈1/2| N ( w )∩ V 0 |⌉ . A function f : V → {0, 1, …, ⌈ Δ /2⌉+1} is a unique response strong Roman function (URStRF), if w ∈ V 0 , then | N ( w )∩ V 2 |≤1 and w ∈ V 1 ∪ V 2 implies that | N ( w )∩ V 2 |=0 . A function f : V → {0, 1, …, ⌈ Δ /2⌉+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 S t R ( 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 S t R ( T )≤8 n /9 .