TREES WITH EQUAL STRONG ROMAN DOMINATION NUMBER AND ROMAN DOMINATION NUMBER

A graph theoretical model called Roman domination in graphs originates from the historical background that any undefended place (with no legions) of the Roman Empire must be protected by a stronger neighbor place (having two legions). It is applicable to military and commercial decision-making problems. A Roman dominating function for a graph G = (V, E) is a function $f:V{\rightarrow}\{0,1,2\}$ such that every vertex v with f(v)=0 has at least a neighbor w in G for which f(w)=2. The Roman domination number of a graph is the minimum weight ${\sum}_{v{\in}V}\;f(v)$ of a Roman dominating function. In order to deal a problem of a Roman domination-type defensive strategy under multiple simultaneous attacks, ${\acute{A}}lvarez$-Ruiz et al. [1] initiated the study of a new parameter related to Roman dominating function, which is called strong Roman domination. ${\acute{A}}lvarez$-Ruiz et al. posed the following problem: Characterize the graphs G with equal strong Roman domination number and Roman domination number. In this paper, we construct a family of trees. We prove that for a tree, its strong Roman dominance number and Roman dominance number are equal if and only if the tree belongs to this family of trees.