Erdös–Gallai-type problems for distance-edge-monitoring numbers

Abstract

Foucaud et al. recently introduced and initiated the study of a new graph-theoretic concept in the area of network monitoring. For every edge e of G and a set M⊆V(G), M is a distance-edge-monitoring (DEM for short) set if there are a vertex x of M and a vertex y of G such that e belongs to all shortest paths between x and y. The DEM numberdem(G) is the smallest size of such a set in G. The vertices of M represent distance probes in a network modeled by G; when the edge e fails, the distance from x to y increases, and thus we are able to detect the failure. In this paper, we study Erdös–Gallai-type problems for DEM numbers of general graphs. The exact values or bounds of dem(G) for radix n-triangular mesh networks and hexagonal networks are also given.