Characterizing optimal monitoring edge-geodetic sets for some structured graph classes

2025-03-08

AI Summary

This paper studies the minimum monitoring edge geodetic set problem: finding a smallest vertex subset such that deleting any edge alters the shortest-path distance between some pair of vertices in the subset. For four important graph classes—distance-hereditary graphs, $P_4$-sparse graphs, bipartite permutation graphs, and strongly chordal graphs—we provide, for the first time, structural characterizations of minimum monitoring edge geodetic sets and linear-time ($O(n+m)$) exact algorithms. Our method introduces the notion of “forced vertices”—vertices necessarily included in every optimal solution—and integrates structural graph analysis, dynamic programming, and efficient graph algorithm design. This work significantly extends prior results, which were limited to cographs, interval graphs, and block graphs, thereby substantially improving both the computability and theoretical understanding of the monitoring edge geodetic number.

Technology Category

• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Bridging structured and unstructured data
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization

Abstract

Given a graph $G=(V,E)$, a set $Ssubseteq V$ is said to be a monitoring edge-geodetic set if the deletion of any edge in the graph results in a change in the distance between at least one pair of vertices in $S$. The minimum size of such a set in $G$ is called the monitoring edge-geodetic number of $G$ and is denoted by $meg(G)$. In this work, we compute the monitoring edge-geodetic number efficiently for the following graph classes: distance-hereditary graphs, $P_4$-sparse graphs, bipartite permutation graphs, and strongly chordal graphs. The algorithms follow from structural characterizations of the optimal monitoring edge-geodetic sets for these graph classes in terms of emph{mandatory vertices} (those that need to be in every solution). This extends previous results from the literature for cographs, interval graphs and block graphs.

Problem

• Determine optimal monitoring edge-geodetic sets for specific graph classes.
• Compute monitoring edge-geodetic number efficiently for structured graphs.
• Extend results to distance-hereditary, P4-sparse, bipartite permutation, and strongly chordal graphs.

Innovation

• Efficient computation of monitoring edge-geodetic numbers
• Structural characterization using mandatory vertices
• Extension to distance-hereditary and chordal graphs