Total Weak Roman Domination in Graphs

Abstract

Given a graph G = ( V , E ) , a function f : V → { 0 , 1 , 2 , ⋯ } is said to be a total dominating function if ∑ u ∈ N ( v ) f ( u ) > 0 for every v ∈ V , where N ( v ) denotes the open neighbourhood of v. Let V i = { x ∈ V : f ( x ) = i } . We say that a function f : V → { 0 , 1 , 2 } is a total weak Roman dominating function if f is a total dominating function and for every vertex v ∈ V 0 there exists u ∈ N ( v ) ∩ ( V 1 ∪ V 2 ) such that the function f ′ , defined by f ′ ( v ) = 1 , f ′ ( u ) = f ( u ) - 1 and f ′ ( x ) = f ( x ) whenever x ∈ V ∖ { u , v } , is a total dominating function as well. The weight of a function f is defined to be w ( f ) = ∑ v ∈ V f ( v ) . In this article, we introduce the study of the total weak Roman domination number of a graph G, denoted by γ t r ( G ) , which is defined to be the minimum weight among all total weak Roman dominating functions on G. We show the close relationship that exists between this novel parameter and other domination parameters of a graph. Furthermore, we obtain general bounds on γ t r ( G ) and, for some particular families of graphs, we obtain closed formulae. Finally, we show that the problem of computing the total weak Roman domination number of a graph is NP-hard.