Total Exact Domination in Graphs

Abstract

Let \( G = (V(G), E(G)) \) be a simple nontrivial undirected graph. A set $T\subseteq V(G)$ is said to be a total exact dominating set if $T$ is both a total dominating set and an exact dominating set of $G$. The cardinality of a minimum total exact dominating set is called the total exact domination number and is denoted by \( \gamma_{te}(G) \). The total exact domination numbers of various types of special graphs, such as path, cycle, star, complete bipartite, and graphs resulting from binary operations such as join, corona, and lexicographic product are obtained in this study. Furthermore, if $G$ does not have a total exact dominating set, then $G$ is called a $non-\gamma_{te}$-graph. Examples include complete graphs, fan graphs, and wheel graphs with more than two vertices. We also consider some disconnected graphs in the corona and lexicographic product, making the study more interesting. In defining total exact domination, a condition of exact domination was modified because it contradicted the definition of total domination.