On the minimal forts of trees
Thomas R. Cameron, Kelvin Li

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…
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Taxonomy
TopicsAdvanced Graph Theory Research · Complexity and Algorithms in Graphs · Interconnection Networks and Systems
