SOME RESULTS ON THE GLOBAL TRIPLE ROMAN DOMINATION IN GRAPHS

A triple Roman dominating function (TRDF) on a graph G with vertex with f (v) < 3, E set V is a function f : V-+ {0, 1, 2, 3, 4} } such that for any vertex v E V x is an element of N(v)boolean OR{v} is an element of N(v) boolean OR{ v } f(x) > |{x E N(v) : f(x) > 1}| }| + 3, where N(v) is the open neighborhood of v. The weight of a TRDF f is the value E v is an element of V is an element of V f (v). A global triple Roman dominating function (GTRDF) on G is a TRDF on both G and its complement. The minimum weight of a GTRDF on G is called the global triple Roman domination number gamma g[3R] (G) of G. We first show that for any tree T on n > 5 vertices, gamma g[3R] (T) < 7n/4 and characterize all extremal trees. We also show that for any graph G on n vertices, gamma g[3R] (G) not equal 3n - 3, and further characterize all graphs G with gamma g[3R] (G) = 3n - k for each k E {4, 5,6, 7}, }, which improves the results given by Nahani Pour et al. (2022).