Monitoring Edge-Geodetic Sets in Graphs

We introduce a new graph-theoretic concept in the area of network monitoring. In this area, one wishes to monitor vertices and/or edges (viewed as graph) order detect and prevent failures. Inspired by two notions studied literature (edge-geodetic sets distance-edge-monitoring sets), we define notion monitoring edge-geodetic set (MEG-set for short) graph G an $$S\subseteq V(G)$$ (that is, every edge lies on some shortest path between S) with additional property that e G, there is vertex pair x, y S such all paths x y. The motivation that, if removed from (for example it ceases function), probes will failure since distance them increase. explore MEG-sets deriving minimum size MEG-set basic classes (trees, cycles, unicyclic graphs, complete grids, hypercubes, ...) prove upper bound using feedback graph.