Bounds on the connected forcing number of a graph

TL;DR

This paper investigates the connected forcing number of graphs, establishing precise bounds based on graph properties like degrees, girth, and order, advancing understanding of zero forcing sets with connectivity constraints.

Contribution

It provides sharp upper and lower bounds on the connected forcing number using key graph parameters, a novel contribution to graph theory.

Findings

Derived bounds based on minimum and maximum degree

Bounds involving girth and order of the graph

Enhanced understanding of zero forcing sets with connectivity constraints

Abstract

In this paper, we study (zero) forcing sets which induce connected subgraphs of a graph. The minimum cardinality of such a set is called the connected forcing number of the graph. We provide sharp upper and lower bounds on the connected forcing number in terms of the minimum degree, maximum degree, girth, and order of the graph.