Dom-forcing sets in graphs

TL;DR

This paper introduces the concept of dom-forcing sets in graphs, exploring their properties and determining the dom-forcing number for specific well-known graphs, thereby extending the understanding of zero forcing and dominating sets.

Contribution

It defines the dom-forcing set, introduces the dom-forcing number, and provides exact values for this number for certain well-known graphs, advancing graph theory knowledge.

Findings

Defined the concept of dom-forcing sets.

Determined the dom-forcing number for specific graphs.

Extended understanding of zero forcing and dominating sets.

Abstract

A dominating set $D_{f}\subseteq V(G)$ of vertices in a graph $G$ is called a \emph{dom-forcing set} if the sub-graph induced by $\langle D_{f} \rangle$ must form a zero forcing set. The minimum cardinality of such a set is known as the dom-forcing number of the graph $G$, denoted by $F_{d}(G)$. This article embarks on an exploration of the dom-forcing number of a graph $G$. Additionally, it delves into the precise determination of $F_{d}(G)$ for certain well-known graphs