Global total Roman domination in graphs

A total Roman dominating function (TRDF) on a graph [Formula: see text] is a function [Formula: see text] satisfying the conditions (i) every vertex [Formula: see text] for which [Formula: see text] is adjacent at least one vertex [Formula: see text] for which [Formula: see text] and (ii) the subgraph of [Formula: see text] induced by the set of all vertices of positive weight has no isolated vertex. The weight of a TRDF is the sum of its function values over all vertices. A total Roman dominating function [Formula: see text] is called a global total Roman dominating function (GTRDF) if [Formula: see text] is also a TRDF of the complement [Formula: see text] of [Formula: see text]. The global total Roman domination number of [Formula: see text] is the minimum weight of a GTRDF on [Formula: see text]. In this paper, we initiate the study of global total Roman domination number and investigate its basic properties. In particular, we relate the global total Roman domination and the total Roman domination and the global Roman domination number.