On the Monitoring-Edge-Geodetic Numbers of Line Graphs.

For a vertex set M , we say that M is a monitoring-edge-geodetic set (MEG-set for short) of graph G , that is, some vertices of M 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 meg (G) of a graph G is defined as the minimum cardinality of a monitoring-edge-geodetic set of G. The line graph L (G) of G is the graph whose vertices are in one-to-one correspondence with the edges of G , that is, if two vertices are adjacent in L (G) if and only if the corresponding edges have a common vertex in G. In this paper, we study the relation between meg (G) and meg (L (G)) , and prove that 2 ≤ meg (L (G)) ≤ | E (G) |. Next, we have determined the exact values for a MEG-set of some special graphs and their line graphs. For a graph G and its line graph L (G) , we prove that meg (L (G)) − meg (G) can be arbitrarily large. [ABSTRACT FROM AUTHOR]