On the Monitoring-Edge-Geodetic Numbers of Line Graphs

Abstract

For a vertex set [Formula: see text], we say that [Formula: see text] is a monitoring-edge-geodetic set (MEG-set for short) of graph [Formula: see text], that is, some vertices of [Formula: see text] can monitor an edge of the graph, if and only if we can remove that edge would change the distance between some pair of vertices in the set. The monitoring-edge-geodetic number [Formula: see text] of a graph [Formula: see text] is defined as the minimum cardinality of a monitoring-edge-geodetic set of [Formula: see text]. The line graph [Formula: see text] of [Formula: see text] is the graph whose vertices are in one-to-one correspondence with the edges of [Formula: see text], that is, if two vertices are adjacent in [Formula: see text] if and only if the corresponding edges have a common vertex in [Formula: see text]. In this paper, we study the relation between [Formula: see text] and [Formula: see text], and prove that [Formula: see text]. Next, we have determined the exact values for a MEG-set of some special graphs and their line graphs. For a graph [Formula: see text] and its line graph [Formula: see text], we prove that [Formula: see text] can be arbitrarily large.