| Outer-independent Roman domination on graphs originates from the defensive strategy of Ancient Rome, which is that if any city without an army is attacked, a neighboring city with two armies could mobilize an army to support it and any two cities have no army can not be adjacent. The outer-independent Roman domination on graphs is an attractive topic in graph theory, and the definition is described as follows. Given a graph G = (V, E), a function f: V (G)→{0,1,2} is an outer-independent Roman dominating function (OIRDF) if f satisfies that every vertex v∈V with f(v) = 0 has at least one adjacent vertex u∈N(v) with f(u) = 2,
where N(v) is the open neighborhood of v, and the set V0 ={v|f(v) = 0} is an independent set. The weight of an OIRDF f is w(f) = Σv∈V f(v). The value of minfw(f) is the outer-independent Roman domination number of G, denoted as γoiR(G). This paper is devoted to the study of the outer-independent Roman domination number of the Cartesian product of paths Pn□Pm. With the help of computer, we find some recursive OIRDFs and then we get the upper bound of γoiR(Pn□Pm). Furthermore, we prove the lower bound of γoiR(Pn□Pm) (n ≤ 3) is equal to its upper bound. Hence, we achieve the exact value of γoiR(Pn□Pm) for n ≤ 3 and the
upper bound of γoiR(Pn□Pm) for n ≥ 4. |
