On the forts and related parameters of the hypercube graph

TL;DR

This paper characterizes minimum forts in hypercube graphs, relates them to automorphisms, and explores their connection to zero forcing parameters, providing new bounds and constructions for these graph invariants.

Contribution

It provides a complete characterization of minimum forts in hypercube graphs and links them to automorphisms and zero forcing parameters, including new bounds and constructions.

Findings

Minimum forts in hypercube graphs are automorphic to two specific sets.

Non-automorphic minimum zero forcing sets have distinct propagation times.

When the hypercube's dimension is a power of two, the fort number equals the domination number.

Abstract

In 2018, forts were defined as non-empty subsets of vertices in a graph where no vertex outside the set has exactly one neighbor in the set. Forts have since been used to characterize zero forcing sets, model zero forcing as an integer program, and provide lower bounds on the zero forcing number. In this article, we give a complete characterization of minimum forts in the hypercube graph, showing that they are automorphic to one of two sets. In contrast, non-automorphic minimum zero forcing sets are identified with distinct propagation times. We also derive the fractional zero forcing number and bounds on the fort number of the hypercube. When the hypercube's dimension is a power of two, the fort number and fractional zero forcing number are equal to the domination number, total domination number, and open packing number. Lastly, we present general constructions for minimal forts in the Cartesian product of graphs, reflecting some minimal forts of the hypercube.