Bounds and extremal graphs for monitoring edge-geodetic sets in graphs

Florent Foucaud, Clara Marcille, Zin Mar Myint, R. B. Sandeep, Sagnik Sen, S. Taruni

Abstract: A monitoring edge-geodetic set, or simply an MEG-set, of a graph G is a vertex subset M⊆V(G) such that given any edge e of G, e lies on every shortest u-v path of G, for some u,v∈M. The monitoring edge-geodetic number of G, denoted by meg(G), is the minimum cardinality of such an MEG-set. This notion provides a graph theoretic model of the network monitoring problem.In this article, we compare meg(G) with some other graph theoretic parameters stemming from the network monitoring problem and provide examples of graphs having prescribed values for each of these parameters. We also characterize graphs G that have V(G) as their minimum MEG-set, which settles an open problem due to Foucaud et al. (CALDAM 2023), and prove that some classes of graphs fall within this characterization. We also provide a general upper bound for meg(G) for sparse graphs in terms of their girth, and later refine the upper bound using the chromatic number of G. We examine the change in meg(G) with respect to two fundamental graph operations: clique-sum and subdivisions. In both cases, we provide a lower and an upper bound of the possible amount of changes and provide (almost) tight examples.