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arXiv · 2507.11194

Connected forcing density and related problems

Boris Brimkov·Randy Davila·Houston Schuerger
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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.

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Boris Brimkov, Randy Davila, Houston Schuerger. 2025-07-15. Connected forcing density and related problems. https://arxiv.org/abs/2507.11194

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