Extremal edge general position sets in some graphs

Abstract

A set of edges $X\subseteq E(G)$ of a graph $G$ is an edge general position set if no three edges from $X$ lie on a common shortest path. The edge general position number ${\mathrm{gp}}_{\mathrm{e}}(G)$ of $G$ is the cardinality of a largest edge general position set in $G$. Graphs $G$ with ${\mathrm{gp}}_{\mathrm{e}}(G)=|E(G)|-1$ and with ${\mathrm{gp}}_{\mathrm{e}}(G)=3$ are respectively characterized. Sharp upper and lower bounds on ${\mathrm{gp}}_{\mathrm{e}}(G)$ are proved for block graphs $G$ and exact values are determined for several specific block graphs.