Monitoring Edge-Geodetic Numbers of Generalized Hierarchical Product Graphs

Let [Formula: see text] be a graph with vertex set [Formula: see text] and edge set [Formula: see text]. For two vertices [Formula: see text] and an edge [Formula: see text], we say [Formula: see text] monitor [Formula: see text] if [Formula: see text] belongs to all shortest paths from [Formula: see text] to [Formula: see text]. For a vertex set [Formula: see text], if every edge of [Formula: see text] can be monitored by a pair of vertices in [Formula: see text], then [Formula: see text] is a monitoring edge-geodetic set (MEG-set for short) of [Formula: see text]. The monitoring edge-geodetic number (MEG number for short), denoted by meg([Formula: see text]), is the size of the smallest MEG-set of [Formula: see text]. In this paper, we obtain exact values or sharp bounds on MEG numbers of generalized hierarchical product graphs.