Dominating the direct product of two graphs through total Roman\n strategies

2020/05/27 by Abel Cabrera Martínez, Martinez, Abel Cabrera, Dorota Kuziak +5
Computer Science · #Advanced Graph Theory Research #Combinatorics (math.CO) #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2005.13608

openalex publication_date 2020/05/27 · openalex created_date 2022/07/19 · openalex updated_date 2026/07/28

Abstract

Given a graph G without isolated vertices, a total Roman dominating\nfunction for G is a function f : V(G)\→ 0,1,2 such that every\nvertex with label 0 is adjacent to a vertex with label 2, and the set of\nvertices with positive labels induces a graph of minimum degree at least one.\nThe total Roman domination number \γ_tR(G) of G is the smallest\npossible value of \∑_{v\∈ V(G)}f(v) among all total Roman dominating\nfunctions f. The total Roman domination number of the direct product G\×\nH of the graphs G and H is studied in this work. Specifically, several\nrelationships, in the shape of upper and lower bounds, between\n\γ_tR(G\× H) and some classical domination parameters for the\nfactors are given. Characterizations of the direct product graphs G\× H\nachieving small values (\≤ 7) for \γ_tR(G\× H) are presented,\nand exact values for \γ_tR(G\× H) are deduced, while considering\nvarious specific direct product classes.\n

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