Counting the number of weakly connected dominating sets of graphs

TL;DR

This paper investigates the enumeration of weakly connected dominating sets in various graphs, providing insights into their structural properties and counting methods.

Contribution

It introduces methods to count weakly connected dominating sets in different classes of graphs, advancing understanding of their combinatorial characteristics.

Findings

Derived formulas for counting weakly connected dominating sets in specific graph classes

Identified structural properties influencing the number of such sets

Provided bounds and exact counts for certain graphs

Abstract

Let $G=(V(G),E(G))$ be a simple graph. A non-empty set $S\subseteq V (G)$ is a weakly connected dominating set in $G$, if the subgraph obtained from $G$ by removing all edges each joining any two vertices in $V (G)\setminus S$ is connected. In this paper, we consider some graphs and study the number of their weakly connected dominating sets.