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

2024-03-14

arXiv.org

AI Summary

This paper investigates the monitoring edge geodetic number (MEG number) of graphs—the minimum cardinality of a vertex set such that every edge lies on some shortest path between two vertices in the set—modeling network link monitoring. Using graph-theoretic analysis, extremal combinatorics, and constructive proofs, we fully characterize the class of graphs whose MEG number equals the number of vertices, resolving an open problem posed by Foucaud et al. at CALDAM 2023. We establish tight upper bounds for the MEG number in terms of girth and chromatic number. We systematically analyze how clique expansions and edge subdivisions affect the MEG number. Furthermore, we derive exact relationships between the MEG number and classical parameters—including domination number and geodetic number—and provide optimal upper bounds for sparse graphs, accompanied by extremal constructions achieving nearly all tight bounds.

Technology Category

• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Machine Learning: Graph-based Machine Learning
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Social Networks and Social Media: Social media analysis through the lenses of networks

Abstract

A monitoring edge-geodetic set, or simply an MEG-set, of a graph $G$ is a vertex subset $M subseteq V(G)$ such that given any edge $e$ of $G$, $e$ lies on every shortest $u$-$v$ path of $G$, for some $u,v in M$. 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 extit{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.

Problem

• Monitoring Edge Geodetic Set
• Graph Theory
• Network Surveillance

Innovation

• Monitoring Edge Geodetic Sets
• Graph Theory Applications
• Bound Formulas