Dominating the direct product of two graphs through total Roman strategies

Identifier:  imarina:8505358

Handle: https://hdl.handle.net/20.500.11797/imarina8505358

Authors:  Cabrera A; Kuziak D; Peterin I; Yero IG

Abstract:
© 2020 by the authors. Given a graph G without isolated vertices, a total Roman dominating function for G is a function f: V(G) → [0, 1, 2] such that every vertex u with f (u) = 0 is adjacent to a vertex v with f (v) = 2, and the set of vertices with positive labels induces a graph of minimum degree at least one. The total Roman domination number γtR(G) of G is the smallest possible value of ΣvεV(G) f (v) among all total Roman dominating functions f . The total Roman domination number of the direct product G× H of the graphs G and H is studied in this work. Specifically, several relationships, in the shape of upper and lower bounds, between γtR(G× H) and some classical domination parameters for the factors are given. Characterizations of the direct product graphs G× H achieving small values (≤ 7) for γtR(G× H) are presented, and exact values for γtR(G× H) are deduced, while considering various specific direct product classes.