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On four network monitoring parameters in graphs and their gaps

Zin Mar Myint, Avikal Srivastava

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.

On four network monitoring parameters in graphs and their gaps

Abstract

Let 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 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 satisfying , does there exist a graph 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.

Paper Structure

This paper contains 10 sections, 14 theorems, 43 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Our contributions
  3. Preliminaries and known results
  4. Main Results
  5. Eliminating impossible parameter quadruples
  6. Realizing the case g(G)=2 and eg(G)=3
  7. General realization for g(G)=2 and eg(G)=3
  8. Generalizing the construction to arbitrary parameters
  9. Complexity of the construction
  10. Conclusion

Key Result

Lemma 1

Let $G$ be a graph with a cut-vertex $u$. Then $u$ is never part of any minimal MEG-set of $G$.

Figures (4)

  • Figure 1: The graph $G_{2,3,c,d}$ for $c\ge 4$. (The vertex $v'_{1(r-1)}$ and the dotted edges are added when $d-c \ge 2$ is even.)
  • Figure 2: $G_{a,b,c,d}$ showing the clique $W$ and extension $V$ when $d-c$ is odd.
  • Figure 3: $G_{a,b,c,d}$ showing the clique $W$ and extension $V$ when $d-c$ is even.
  • Figure :

Theorems & Definitions (23)

  • Lemma 1: foucaud2023monitoring
  • Lemma 2: foucaud2023monitoring
  • Lemma 3: foucaud2023monitoring
  • Proposition 4: foucaud2025
  • Theorem 5: foucaud2025
  • Theorem 6
  • proof
  • Theorem 7
  • proof
  • Lemma 8
  • ...and 13 more