On the total domination number of total graphs

Abstract

Let G be a graph with no isolated vertex. A set D ⊆ V (G) is a total dominating set of G if every vertex of G is adjacent to at least one vertex in D. The total domination number of G, denoted by γ t (G), is the minimum cardinality among all total dominating sets of G. In this paper we study the total domination number of total graphs T(G) of simple graphs G. In particular, we give some relationships that exist between γ t (T(G)) and other domination parameters of G and of some well-known graph operators on G. Finally, we provide closed formulas on γ t (T(G)) for some well-known families of graphs G.

Some Characterizations of Total Outer-connected Domination in Graphs

“…Modeling and analyzing various real-world systems, such as social networks, transportation networks, and communication networks, is facilitated by graphs (Saromines & Canoy Jr, 2022). The study of domination, which seeks to determine the minimum number of vertices required to control or influence the entire graph, is one of the fundamental problems in graph theory Cabrera-Martínez et al (2023). Total outer-connected domination is a variant of the traditional domination problem that emphasizes dominating the entire graph while taking into account the connectivity between vertices (Senthilkumar et al, 2023).…”