Zero forcing number versus general position number in tree-like graphs

TL;DR

This paper investigates the relationship between zero forcing number and general position number in various tree-like graphs, extending known inequalities to new classes and identifying cases where the inequalities do not hold.

Contribution

It extends the inequality ${ m gp}(G) geq { m Z}(G)$ from trees to block graphs and certain quasi-trees, and explores its limitations in bicyclic graphs.

Findings

${ m gp}(T) geq { m Z}(T) + 1$ for trees

${ m gp}(G) geq { m Z}(G)$ for connected unicyclic graphs

The inequality does not extend to bicyclic graphs or all quasi-trees

Abstract

Let ${\rm Z}(G)$ and ${\rm gp}(G)$ be the zero forcing number and the general position number of a graph $G$, respectively. Known results imply that ${\rm gp}(T)\ge {\rm Z}(T) + 1$ holds for every nontrivial tree $T$. It is proved that the result extends to block graphs. For connected, unicyclic graphs $G$ it is proved that ${\rm gp}(G) \ge {\rm Z}(G)$. The result extends neither to bicyclic graphs nor to quasi-trees. Nevertheless, a large class of quasi-trees is found for which ${\rm gp}(G) \ge {\rm Z}(G)$ holds.