Global triple Roman dominating function

Abstract

Let k be a positive integer and G=(V,E) a graph. A [k]-Roman dominating function is a function f:V→{0,1,2,…,k+1} such that for every v∈V(G) with f(v)<k, f(AN[v])≥|AN(v)|+k, where AN(v) is the set of neighbors of v assigned a non-zero value under f and AN[v]=AN(v)∪{v}. When k=3, the function f is called a triple Roman dominating function (TRD-function). A global triple Roman dominating function (GTRD-function) on a graph G=(V,E) is a TRD-function for both G and its complement graph G¯. The global triple Roman domination number of a graph G is the minimum weight over all GTRD-functions on G. In this paper, we start the study of the global triple Roman domination number and we obtain various tight bounds for it. Moreover, we prove that the global triple Roman domination problem is NP-complete for bipartite and chordal graphs.