On four network monitoring parameters in graphs and their gaps

TL;DR

This paper characterizes which quadruples of four network monitoring parameters can be realized by connected graphs, providing explicit constructions for all admissible cases and identifying impossible ones.

Contribution

It offers a complete, constructive characterization of realizable quadruples of network monitoring parameters in connected graphs, including non-realizable cases.

Findings

Identified quadruples of parameters that cannot be realized by any connected graph.

Constructed explicit graphs for all admissible quadruples with linear growth in vertices and edges.

Provided a complete classification of realizable and non-realizable parameter quadruples.

Abstract

Let \( G \) be a finite simple undirected graph. Four graph parameters related to network monitoring are the \emph{geodetic set}, \emph{edge geodetic set}, \emph{strong edge geodetic set}, and \emph{monitoring edge geodetic set}, with corresponding minimum sizes, denoted by \( g(G), eg(G), seg(G) \), and \( meg(G) \), respectively. These parameters quantify the minimum number of vertices required to monitor all vertices and edges of \( G \) under progressively stricter path-based conditions. As established by Florent \textit{et al.}\ (CALDAM 2023), these parameters satisfy the chain of inequalities: \( g(G) \leq eg(G) \leq seg(G) \leq meg(G). \) In 2025, Florent \textit{et al.}\ posed the following question: given integers \( a, b, c, d \) satisfying \( 2 \leq a \leq b \leq c \leq d \), does there exist a graph \( G \) such that \( g(G) = a, \quad eg(G) = b, \quad seg(G) = c, \quad \text{and} \quad meg(G) = d? \) They partially answered this affirmatively under three specific hypotheses and gave some constructions to support it. In this article, we first identify quadruples of values that cannot be realized by any connected graph. For all remaining admissible quadruples, we provide explicit constructions of connected graphs that realize the specified parameters. These constructions are modular and efficient, with the number of vertices and edges growing linearly with the largest parameter, providing a complete and constructive characterization of such realizable quadruples.