Some results on minimum skew zero forcing sets, and skew zero forcing number

TL;DR

This paper investigates the properties of skew zero forcing sets in graphs, characterizes certain graph classes based on their skew zero forcing number, and explores relationships with matchings and matroids.

Contribution

It provides characterizations of graphs with extreme skew zero forcing numbers and links minimum skew zero forcing sets to matroid bases in bipartite graphs.

Findings

Characterization of complete multipartite graphs by skew zero forcing number

Relation between minimum skew zero forcing sets and matchings in bipartite and unicyclic graphs

Minimum skew zero forcing sets form matroid bases in certain bipartite graphs

Abstract

Let $G$ be a graph, and $Z$ a subset of its vertices, which we color black, while the remaining are colored white. We define the skew color change rule as follows: if $u$ is a vertex of $G$, and exactly one of its neighbors $v$, is white, then change the color of $v$ to black. A set $Z$ is a skew zero forcing set for $G$ if the application of the skew color change rule (as many times as necessary) will result in all the vertices in $G$ colored black. A set $Z$ is a minimum skew zero forcing set for $G$ if it is a skew zero forcing set for $G$ of least cardinality. The skew zero forcing number $\sZ (G)$ is the minimum of $|Z|$ over all skew zero forcing sets $Z$ for $G$. In this paper we discuss graphs that have extreme skew zero forcing number. We characterize complete multipartite graphs in terms of $\sZ (G)$. We note relations between minimum skew zero forcing sets and matchings in some bipartite graphs, and in unicyclic graphs. We establish that the elements in the set of minimum skew zero forcing sets in certain bipartite graphs are the bases of a matroid.