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

Abstract

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