weak signed roman domination in graphs

A weak signed roman dominating function (wsrdf) of a graph g with vertex set v(g) is defined as a function f:v(g)→{−1,1,2} having the property that ∑x∈n[v]f(x)≥1 for each v∈v(g), where n[v] is the closed neighborhood of v. the weight of a wsrdf is the sum of its function values over all vertices. the weak signed roman domination number of g, denoted by γwsr(g), is the minimum weight of a wsrdf in g. we initiate the study of the weak signed roman domination number, and we present different sharp bounds on γwsr(g). in addition, we determine the weak signed roman domination number of some classes of graphs