On the minimal forts of trees

TL;DR

This paper studies minimal forts in trees, providing a combinatorial characterization, bounds on their number, and identifying trees that attain these bounds, linking to parameters like star centers and zero forcing number.

Contribution

It introduces a combinatorial-cut characterization of minimal forts in trees and establishes bounds and characterizations related to these forts.

Findings

Derived an upper bound on the size of minimal forts in trees.

Established a lower bound on the number of minimal forts in trees.

Characterized trees that achieve the lower bound through a four-part equivalence theorem.

Abstract

In 2018, the concept of a fort in graph theory was introduced as a non-empty subset of vertices satisfying the condition that no vertex outside the set has exactly one neighbor in the set. Since then, forts have played a significant role in characterizing zero forcing sets, modeling the zero forcing number as an integer program, and generating lower bounds for the zero forcing number of Cartesian products. Recent research has focused on the number of minimal forts, defined as those for which no proper subset is a fort. Notably, it has been established that the number of minimal forts in any graph is strictly less than Sperner's bound, a famous bound due to Emanuel Sperner (1928) on the size of a collection of subsets where no subset contains another. Moreover, lower bounds on the number of minimal forts for several families of graphs were established, and it was shown that certain families have an exponential number of minimal forts. In this article, we provide a combinatorial-cut characterization of the minimal forts in trees. Using this characterization, we derive an upper bound on the cardinality of minimal forts and a lower bound on the number of minimal forts in trees. We also characterize the trees that attain this lower bound through a four-part equivalence theorem that provides a connection to other graph parameters, such as star centers, the fort number, and the zero forcing number.