Characterization of graphs with orientable total domination number equal to $|V|-1$

TL;DR

This paper characterizes graphs whose valid orientations have a total domination number equal to one less than the number of vertices, and explores the extremal values of orientable total domination numbers.

Contribution

It provides a characterization of graphs with maximum orientable total domination number of |V|-1 and analyzes the range of possible total domination numbers across orientations.

Findings

Graphs with $ ext{DOM}_t(G) = |V|-1$ are characterized.

Existence of graph families with maximum and minimum orientable total domination numbers.

$ ext{DOM}_t(G)$ and $ ext{dom}_t(G)$ can differ significantly, with $ ext{DOM}_t(G)=|V|-1$ and $ ext{dom}_t(G)=3$.

Abstract

In a directed graph $D$, a vertex subset $S\subseteq V$ is a total dominating set if every vertex of $D$ has an in-neighbor from $S$. A total dominating set exists if and only if every vertex has at least one in-neighbor. We call the orientation of such directed graphs valid. The total domination number of $D$, denoted by $\gamma_t(D)$, is the size of the smallest total dominating set of $D$. For an undirected graph $G$, we investigate the upper (or lower) orientable total domination number of $G$, denoted by $\mathrm{DOM}_t(G)$ (or $\mathrm{dom}_t(G)$), that is the maximum (or minimum) of the total domination numbers over all valid orientations of $G$. We characterize those graphs for which $\mathrm{DOM}_t(G)=|V(G)|-1$, and consequently we show that there exists a family of graphs for which $\mathrm{DOM}_t(G)$ and $\mathrm{dom}_t(G)$ can be as far as possible, namely $\mathrm{DOM}_t(G)=|V(G)|-1$ and $\mathrm{dom}_t(G)=3$.