A New Upper Bound on Total Domination Number of Bipartite Graphs

TL;DR

This paper establishes a new upper bound on the total domination number of bipartite graphs with minimum degree at least k, using a greedy algorithm, providing a tighter estimate than previous bounds.

Contribution

The paper introduces a novel upper bound for the total domination number of bipartite graphs with minimum degree k, derived via a greedy algorithm, extending existing theoretical results.

Findings

Derived a new upper bound for total domination number in bipartite graphs

The bound depends on the order of the graph and the minimum degree k

The result applies to graphs with minimum degree greater than 1

Abstract

Let $ G $ be a graph. A subset $S \subseteq V(G) $ is called a total dominating set if every vertex of $G$ is adjacent to at least one vertex of $S$. The total domination number, $\gamma_{t}$($G$), is the minimum cardinality of a total dominating set of $G$. In this paper using a greedy algorithm we provide an upper bound for $\gamma_{t}$($G$), whenever $G$ is a bipartite graph and $\delta(G)$ $\geq$ $k$. More precisely, we show that if $k$ > 1 is a natural number, then for every bipartite graph $G$ of order $n$ and $\delta(G) \ge k$, $ $$\gamma_{t}$($G$) $\leq$ $n(1- \frac{k!}{\prod_{i=0}^{k-1}(\frac{k}{k-1}+i)}).$