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Reconfiguration of Minimum PSD Forcing Sets and Minimum Skew Forcing Sets

Novi Bong, Mary Flagg, Mark Hunnell, John Hutchens, Ryan Moruzzi, Houston Schuerger, Ben Small

TL;DR

This work develops a universal framework for TE/TS reconfiguration graphs applied to PSD and skew forcing variants, enabling broad conclusions about structure, realizability, and inter-rule comparisons. It formalizes relaxations such as relaxed chronologies and path bundles, and classifies parameters into summable and coverable to derive wide-ranging results, including disjointness structures and clique bounds. The authors provide concrete characterizations for PSD and skew forcing on trees and complete graphs, demonstrate how PSD and skew reconfiguration graphs can diverge or align, and reveal deep connections such as the SD$_2$ distance-2 graph correspondence for skew-nontrivial trees. The work also contrasts reconfiguration behavior across color-change rules and discusses realizability and component structure for various graph families, highlighting the nuanced dependence on the forcing variant and source graph.

Abstract

Reconfiguration graphs provide a way to represent relationships among solutions to a problem, and have been studied in many contexts. We investigate the reconfiguration graphs corresponding to minimum PSD forcing sets and minimum skew forcing sets. We present results for the structure and realizability of certain graph classes as token exchange and token sliding reconfiguration graphs. Additionally, we use a universal approach to establish structural properties for the reconfiguration graphs of many common graph parameters under these reconfiguration rules. Finally, we compare results on reconfiguration graphs for zero forcing variants.

Reconfiguration of Minimum PSD Forcing Sets and Minimum Skew Forcing Sets

TL;DR

This work develops a universal framework for TE/TS reconfiguration graphs applied to PSD and skew forcing variants, enabling broad conclusions about structure, realizability, and inter-rule comparisons. It formalizes relaxations such as relaxed chronologies and path bundles, and classifies parameters into summable and coverable to derive wide-ranging results, including disjointness structures and clique bounds. The authors provide concrete characterizations for PSD and skew forcing on trees and complete graphs, demonstrate how PSD and skew reconfiguration graphs can diverge or align, and reveal deep connections such as the SD distance-2 graph correspondence for skew-nontrivial trees. The work also contrasts reconfiguration behavior across color-change rules and discusses realizability and component structure for various graph families, highlighting the nuanced dependence on the forcing variant and source graph.

Abstract

Reconfiguration graphs provide a way to represent relationships among solutions to a problem, and have been studied in many contexts. We investigate the reconfiguration graphs corresponding to minimum PSD forcing sets and minimum skew forcing sets. We present results for the structure and realizability of certain graph classes as token exchange and token sliding reconfiguration graphs. Additionally, we use a universal approach to establish structural properties for the reconfiguration graphs of many common graph parameters under these reconfiguration rules. Finally, we compare results on reconfiguration graphs for zero forcing variants.
Paper Structure (29 sections, 74 theorems, 32 equations, 8 figures, 2 tables)

This paper contains 29 sections, 74 theorems, 32 equations, 8 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries and notation
  3. Graph terminology
  4. Zero forcing
  5. Reconfiguration
  6. Relaxed chronologies and path bundles
  7. Universal Parameter Properties
  8. Parameter classes
  9. Partition parameters
  10. Cliques in reconfiguration graphs
  11. Reconfiguration for Positive Semidefinite Forcing
  12. Reconfiguration of Trees
  13. Reconfiguration of Complete Graphs
  14. Disconnected reconfiguration graphs
  15. Reconfiguration for Skew Forcing
  16. ...and 14 more sections

Key Result

Theorem 2.2.2

Let $G$ be a graph, and $B \subseteq V(G)$. $B$ is a PSD forcing set of $G$ if and only if $B \cap F \neq \emptyset$ for each PSD fort $F$ of $G$.

Figures (8)

  • Figure 3.1: Illustration of set structure in proof of \ref{['SetDifferenceLemma']}.
  • Figure 4.1: Up to symmetry, there are two minimum PSD forcing sets for $K_{4,4,4}$.
  • Figure 5.1: The graph $KB(7,3)$
  • Figure 6.1: A skew-nontrivial tree $T$ and its special distance-2 graph $\mathop{SD_2}(T)$.
  • Figure 7.1: Nonstandard graphs referenced in \ref{['noncomp']}.
  • ...and 3 more figures

Theorems & Definitions (147)

  • Definition 2.1.1
  • Definition 2.1.2
  • Definition 2.1.3
  • Definition 2.2.1: psdfort
  • Theorem 2.2.2: psdfort
  • Definition 2.2.3
  • Lemma 2.2.4
  • proof
  • Definition 2.2.5
  • Proposition 2.2.6
  • ...and 137 more