Abstract
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 ∑v∈V(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.
Highlights
The present investigation is devoted to describe several contributions to the theory of total Roman dominating functions while dealing with the direct product of two graphs.Studies concerning parameters in relation to domination in graphs are very frequently present in recent years
This might probably be caused by the popularity of some classical problems, like for instance
The conjecture claims that the cardinality of the smallest dominating set of the Cartesian product of two graphs is at least equal to the product of the domination numbers of the factor graphs involved in the product
Summary
The present investigation is devoted to describe several contributions to the theory of total Roman dominating functions while dealing with the direct (or tensor or Kronecker) product of two graphs. The parameter of G called Roman domination number stands for the least weight among all functions that are proved to be Roman dominating on G. Is called a total Roman dominating function if V1 ∪ V2 induces a graph without isolated vertices. Roman domination number of G stands for the minimum possible weight among all total Roman dominating functions on G This parameter is denoted γtR ( G ). Roman dominating function whose weight equals precisely γtR ( G ) These concepts of total Roman domination were first introduced in [15] by using some more general settings. The parameter called open packing number of G is the cardinality of the largest possible open packing set of G We write this cardinality by using the notation ρo ( G ).
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