Monitoring Edge-Geodetic Sets in Graphs

Abstract

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

Monitoring edge-geodetic numbers of convex polytopes and four networks

“…In Lemma 2.1. [5] If two vertices are twins of degree at least 1 in graph G, then they must belong to any MEG-set of G. Lemma 2.2. [7] For any n-dimensional BF (n), there is a unique shortest path of length n from (x; 0) to (y; n).…”

“…Definition 1.1. [5] Two vertices x and y monitor an edge e in graph G if e belongs to all shortest paths from x to y. A set S of vertices of G is called a monitoring edgegeodetic set of G ( MEG-set for short) if, for every edge e of G, there is a pair x, y of vertices of S that monitors e. Definition 1.2.…”

“…A set S of vertices of G is called a monitoring edgegeodetic set of G ( MEG-set for short) if, for every edge e of G, there is a pair x, y of vertices of S that monitors e. Definition 1.2. [5] The minimum size of monitoring the edge-geodetic set is called the monitoring edge-geodetic number (meg(G) for short). Obviously, we can select all vertices as the monitoring edge-geodetic set, so the size of this set can be an upper bound.…”