Well-failed graphs

TL;DR

This paper introduces the concept of well-failed graphs, characterizes trees with this property, and explores the relationship between minimal forts and zero forcing sets, advancing understanding of graph structures related to zero forcing.

Contribution

It defines well-failed graphs, characterizes trees that are well-failed, and links minimal forts with zero forcing sets, providing new insights into graph theory.

Findings

Characterization of well-failed trees.

Equivalence of vertices outside minimal forts and zero forcing sets.

Identification of vertices in no minimal fort in trees.

Abstract

In this paper we begin the study of well-failed graphs, that is, graphs in which every maximal failed zero forcing set is a maximum failed zero forcing set, or equivalently, in which every minimal fort is a minimum fort. We characterize trees that are well-failed. Along the way, we prove that the set of vertices in a graph that are not in any minimal fort is identical to the set of vertices that are in no minimal zero forcing set, which allows us to characterize vertices in a tree that are in no minimal fort.