Zero forcing propagation time intervals and graphs with fixed propagation time

TL;DR

This paper investigates the conditions under which all minimal zero forcing sets in a graph share the same propagation time, introducing the concept of fixed propagation time and characterizing such graphs for standard and positive semidefinite forcing rules.

Contribution

It introduces the notion of fixed propagation time in zero forcing, characterizes graphs with this property for two forcing variants, and proves conjectures for joins of graphs.

Findings

Graphs with fixed propagation time equal to one are characterized.

Families of graphs with longer fixed propagation time for standard forcing are identified.

Such graphs do not exist for positive semidefinite forcing.

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.