Connected forcing density and related problems

TL;DR

This paper introduces CF-dense graphs, explores their properties, characterizes CF-dense trees, and examines how CF-density behaves under various graph operations, advancing understanding of connected forcing sets.

Contribution

It defines CF-dense graphs, characterizes CF-dense trees, and analyzes CF-density preservation under graph operations, providing new insights into connected forcing sets.

Findings

CF-dense graphs include several known graph families.

A formula for the number of connected forcing sets in trees is provided.

CF-density is preserved under certain graph operations like Cartesian products.

Abstract

A connected forcing set of a graph is a zero forcing set that induces a connected subgraph. In this paper, we introduce and study CF-dense graphs -- graphs in which every vertex belongs to some minimum connected forcing set. We identify several CF-dense graph families and investigate the relationships between CF-density and analogous notions in zero forcing and total forcing. We also characterize CF-dense trees and give a formula for the number of distinct connected forcing sets in trees. Finally, we analyze when CF-density is preserved under graph operations such as Cartesian products, joins, and coronas.