Minimal Zero Forcing Sets

TL;DR

This paper investigates the properties of minimal zero forcing sets in graphs, analyzing their quantity, maximum size, and relationships with the zero forcing number, revealing surprising invariances and characterizations.

Contribution

It provides new insights into the structure and extremal properties of minimal zero forcing sets, including conditions for their quantity and size, and their behavior under graph modifications.

Findings

Number of minimal zero forcing sets can be polynomial or exponential.

The maximum size of a minimal zero forcing set relates to the zero forcing number.

Equality of maximum size and zero forcing number is preserved by deleting, but not adding, a universal vertex.

Abstract

In this paper, we study minimal (with respect to inclusion) zero forcing sets. We first investigate when a graph can have polynomially or exponentially many distinct minimal zero forcing sets. We also study the maximum size of a minimal zero forcing set $\overline{\operatorname{Z}}(G)$, and relate it to the zero forcing number $\operatorname{Z}(G)$. Surprisingly, we show that the equality $\overline{\operatorname{Z}}(G)=\operatorname{Z}(G)$ is preserved by deleting a universal vertex, but not by adding a universal vertex. We also characterize graphs with extreme values of $\overline{\operatorname{Z}}(G)$ and explore the gap between $\overline{\operatorname{Z}}(G)$ and $\operatorname{Z}(G)$.