Extremal edge general position sets in some graphs

TL;DR

This paper investigates the maximum size of edge sets in graphs where no three edges lie on a common shortest path, characterizes extremal cases, and provides bounds and exact values for specific graph classes.

Contribution

It characterizes graphs with extremal edge general position numbers and establishes bounds for block graphs, advancing understanding of edge configurations in shortest path structures.

Findings

Characterized graphs with ${ m gp}_{ m e}(G) = |E(G)| - 1$ and ${ m gp}_{ m e}(G) = 3$.

Proved sharp bounds on ${ m gp}_{ m e}(G)$ for block graphs.

Determined exact ${ m gp}_{ m e}(G)$ values for specific block 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 ${\rm gp}_{\rm e}(G)$ of $G$ is the cardinality of a largest edge general position set in $G$. Graphs $G$ with ${\rm gp}_{\rm e}(G) = |E(G)| - 1$ and with ${\rm gp}_{\rm e}(G) = 3$ are respectively characterized. Sharp upper and lower bounds on ${\rm gp}_{\rm e}(G)$ are proved for block graphs $G$ and exact values are determined for several specific block graphs.