Signed double roman domination of graphs

Abstract

In this paper we continue the study of signed double Roman dominating functions in graphs. A signed double Roman dominating function (SDRDF) on a graph G = (V,E) is a function f : V(G)  {-1,1,2,3} having the property that for each v  V(G), f [v] ≥ 1, and if f (v)= -1, then vertex v has at least two neighbors assigned 2 under f or one neighbor w with f(w)=3, and if f(v)=1, then vertex v must have at leat one neighbor w with f (w)≥2. The weight of a SDRDF is the sum of its function values over all vertices. The signed double Roman domination number γsdR(G) is the minimum weight of a SDRDF on G. We present several lower bounds on the signed double Roman domination number of a graph in terms of various graph invariants. In particular, we show that if G is a graph of order n and size m with no isolated vertex, then γsdR(G) ≥ 19n-24m/9 and γsdR(G)≥4 √n/3-n. Moreover, we characterize the graphs attaining equality in these two bounds.