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Well-failed graphs

Bonnie Jacob

TL;DR

The paper introduces well-failed graphs, linking maximal failed zero forcing sets, forts, and zero blocking sets within the zero forcing framework. It establishes a fundamental equivalence among maximal stalled sets, maximal failed sets, minimal forts, and minimal zero blocking sets, enabling a complete tree-focused classification of well-failed graphs. The authors prove that leafy trees form a large well-failed family and provide a full characterization of well-failed trees: they are precisely $P_n$ with $n\in\{1,2,3,4,6\}$, the tree $star_{2,2,2}$, or leafy. They further extend the analysis to cycles and several well-known graphs, including $K_n$, $K_{m,n}$, and the Petersen graph, offering structural insight into how forts and zero forcing interact with star-removal procedures. Overall, the results yield a cohesive framework for understanding well-failed behavior across diverse graph families and introduce tools like star removals and pendent generalized stars for forts analysis.

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.

Well-failed graphs

TL;DR

The paper introduces well-failed graphs, linking maximal failed zero forcing sets, forts, and zero blocking sets within the zero forcing framework. It establishes a fundamental equivalence among maximal stalled sets, maximal failed sets, minimal forts, and minimal zero blocking sets, enabling a complete tree-focused classification of well-failed graphs. The authors prove that leafy trees form a large well-failed family and provide a full characterization of well-failed trees: they are precisely with , the tree , or leafy. They further extend the analysis to cycles and several well-known graphs, including , , and the Petersen graph, offering structural insight into how forts and zero forcing interact with star-removal procedures. Overall, the results yield a cohesive framework for understanding well-failed behavior across diverse graph families and introduce tools like star removals and pendent generalized stars for forts analysis.

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.

Paper Structure

This paper contains 6 sections, 28 theorems, 4 figures.

Table of Contents

  1. Introduction
  2. Definitions and notation
  3. Preliminaries and basic results
  4. Irrelevant vertices
  5. Leafy graphs and well-failed trees
  6. Other well-failed graphs

Key Result

Lemma 1.3

The following are equivalent in a graph $G$ for $S \subseteq V(G)$.

Figures (4)

  • Figure 1: A tree $T$ with $B_0$ and $B_1$ star centers shaded black and gray respectively, and the forest $T_1$ after all $B_0$ star removals have been performed on $T$.
  • Figure 2: A leafy graph.
  • Figure 3: The only well-failed tree that has a high-degree vertex and is not leafy, $\operatorname{star}_{2,2,2}$
  • Figure 4: The labeled Petersen graph

Theorems & Definitions (52)

  • Definition 1.1
  • Definition 1.2
  • Lemma 1.3
  • proof
  • Definition 1.4
  • Proposition 1.6
  • Lemma 1.7
  • proof
  • Definition 1.8
  • Lemma 2.1
  • ...and 42 more