Lower bounds for the total (distance) $k$-domination number of a graph

TL;DR

This paper extends known lower bounds on the total domination number to the broader context of total (distance) $k$-domination numbers in graphs, providing generalized theoretical bounds.

Contribution

It generalizes several existing lower bounds from total domination to total (distance) $k$-domination, broadening the theoretical understanding.

Findings

Several known lower bounds on total domination number are generalized to total (distance) $k$-domination.

The results apply to graphs without isolated vertices.

Provides a unified framework for bounds across different $k$ values.

Abstract

For $k \geq 1$ and a graph $G$ without isolated vertices, a \emph{total (distance) $k$-dominating set} of $G$ is a set of vertices $S \subseteq V(G)$ such that every vertex in $G$ is within distance $k$ to some vertex of $S$ other than itself. The \emph{total (distance) $k$-domination number} of $G$ is the minimum cardinality of a total $k$-dominating set in $G$, and is denoted by $\gamma_{k}^t(G)$. When $k=1$, the total $k$-domination number reduces to the \emph{total domination number}, written $\gamma_t(G)$; that is, $\gamma_t(G) = \gamma_{1}^t(G)$. This paper shows that several known lower bounds on the total domination number generalize nicely to lower bounds on total (distance) $k$-domination.