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Zero forcing propagation time intervals and graphs with fixed propagation time

Daniela Ferrero, H. Tracy Hall, Leslie Hogben, Mark Hunnell, Ben Small

TL;DR

This work investigates propagation time in standard and PSD zero forcing, introducing fixed propagation time and the propagation-time interval as central concepts. It identifies two fast-join constructions that yield fixed propagation time equal to one for both variants and conjectures these are the only such graphs, proving the conjectures in the case of joins. The authors demonstrate a PSD counterexample family showing the PSD propagation time interval need not be full, addressing Warnberg's conjecture in the PSD setting. They also show fixed standard propagation time greater than one can occur (e.g., cycles and threshold graphs) while fixed PSD propagation time greater than one is impossible, and they develop join-based tools that characterize when joins have PT1, establishing that joins with PT1 are fast joins under certain conditions. Overall, the paper advances the understanding of propagation-time realizability, fixed propagation times, and structural characterizations of fast-join graphs in both forcing variants.

Abstract

Zero forcing in a graph refers to the evolution of vertex states under repeated application of a color change rule. Typically the states are chosen to be blue and white, and a forcing set is an initial set of blue vertices such that all of the vertices are blue at the end of the process. In this context, the propagation time of a set in a graph is the number of iterations of the color change rule required to have all vertices blue, performing independent color changes simultaneously. Different minimal forcing sets need not have the same propagation time, and we study the realizability of specific integers as propagation times of minimal forcing sets in graphs for two of the most well-studied color change rules (standard and positive semidefinite). Particular attention is paid to the case where all minimal forcing sets have the same propagation time, and we term this phenomenon fixed propagation time. For each of the two variants, we present a general form of graphs all of which have fixed propagation time equal to one. We conjecture that these are the only such graphs and prove the conjectures for joins of graphs. Families of graphs with longer fixed propagation time for standard forcing are exhibited, and it is shown that such graphs do not exist for positive semidefinite forcing.

Zero forcing propagation time intervals and graphs with fixed propagation time

TL;DR

This work investigates propagation time in standard and PSD zero forcing, introducing fixed propagation time and the propagation-time interval as central concepts. It identifies two fast-join constructions that yield fixed propagation time equal to one for both variants and conjectures these are the only such graphs, proving the conjectures in the case of joins. The authors demonstrate a PSD counterexample family showing the PSD propagation time interval need not be full, addressing Warnberg's conjecture in the PSD setting. They also show fixed standard propagation time greater than one can occur (e.g., cycles and threshold graphs) while fixed PSD propagation time greater than one is impossible, and they develop join-based tools that characterize when joins have PT1, establishing that joins with PT1 are fast joins under certain conditions. Overall, the paper advances the understanding of propagation-time realizability, fixed propagation times, and structural characterizations of fast-join graphs in both forcing variants.

Abstract

Zero forcing in a graph refers to the evolution of vertex states under repeated application of a color change rule. Typically the states are chosen to be blue and white, and a forcing set is an initial set of blue vertices such that all of the vertices are blue at the end of the process. In this context, the propagation time of a set in a graph is the number of iterations of the color change rule required to have all vertices blue, performing independent color changes simultaneously. Different minimal forcing sets need not have the same propagation time, and we study the realizability of specific integers as propagation times of minimal forcing sets in graphs for two of the most well-studied color change rules (standard and positive semidefinite). Particular attention is paid to the case where all minimal forcing sets have the same propagation time, and we term this phenomenon fixed propagation time. For each of the two variants, we present a general form of graphs all of which have fixed propagation time equal to one. We conjecture that these are the only such graphs and prove the conjectures for joins of graphs. Families of graphs with longer fixed propagation time for standard forcing are exhibited, and it is shown that such graphs do not exist for positive semidefinite forcing.

Paper Structure

This paper contains 7 sections, 25 theorems, 2 equations, 7 figures.

Table of Contents

  1. Introduction
  2. A graph that does not have a full PSD propagation time interval
  3. Fixed propagation time greater than one
  4. Fixed standard propagation time greater than one
  5. No graph has fixed PSD propagation time greater than one
  6. Fast joins have fixed propagation time equal to one
  7. Joins with fixed propagation time equal to one are fast joins

Key Result

Theorem 1.6

If $G$ is a PSD fast join, then $\overline{\mathop{\mathrm{pt}}\nolimits_+}(G)=1$. If $G$ is a standard fast join, then $\overline{\mathop{\mathrm{pt}}\nolimits}(G)=1$.

Figures (7)

  • Figure 2.1: The graph ${\mathsf S}_{0}$, which does not have a full PSD propagation interval nor a full expanded PSD propagation time interval.
  • Figure 3.1: The connected graphs of order $n=4,5$ with $\underline{\mathop{\mathrm{pt}}\nolimits}(G)=\overline{\mathop{\mathrm{pt}}\nolimits}(G)=2$
  • Figure 3.2: A threshold graph and its construction tree
  • Figure 3.3: The cograph $W_5$ and a construction tree
  • Figure 3.4: A cograph $G$ with $\mathop{\mathrm{Z}}\nolimits(G)\ne \overline\mathop{\mathrm{Z}}\nolimits(G)$ and a construction tree
  • ...and 2 more figures

Theorems & Definitions (54)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Theorem 1.6
  • Conjecture 1.7
  • Conjecture 1.8
  • Theorem 1.9
  • Definition 2.1
  • ...and 44 more