Connected dom-forcing sets in graphs

TL;DR

This paper investigates the connected dom-forcing number in graphs, exploring its properties, calculating it for well-known graphs, and analyzing its behavior in graph products.

Contribution

It introduces the concept of the connected dom-forcing number, studies its properties, and determines its values for various standard graphs and their products.

Findings

Characterized the properties of the connected dom-forcing number.

Calculated Fcd(G) for several well-known graphs.

Analyzed the behavior of Fcd(G) in graph products.

Abstract

In a graph G, a dominating set Df subset of V (G) is called a dom-forcing set if the sub-graph induced by Df 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 Fd(G). A connected dom-forcing forcing set of a graph G, is a dom-forcing set of G that induces a sub graph of G which is connected. The connected dom-forcing number of G, Fcd(G), is the minimum size of a connected dom-forcing set. This study delves into the concept of the connected dom-forcing number Fcd(G), examining its properties and characteristics. Furthermore, it seeks to accurately determine Fcd(G) for several well-known graphs and their graph products.