Extremal edge general position sets in some graphs

Abstract

A set of edges X ⊆ 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 gp e (G) of G is the cardinality of a largest edge general position set in G. Graphs G with gp e (G) = |E(G)| − 1 and with gp e (G) = 3 are respectively characterized. Sharp upper and lower bounds on gp e (G) are proved for block graphs G and exact values are determined for several specific block graphs.

Generalization of edge general position problem

“…The edge k-general position problem is to find an edge k-gp set. The edge 3-general position problem is known as the edge general position problem and was studied for the first time in [23] and then continued in [13,14]. The corresponding invariant is called the gp e -number of G and denoted by gp e (G).…”