Complexity of Edge Monitoring on Some Graph Classes

TL;DR

This paper investigates the computational complexity of the edge monitoring problem across various graph classes, proposing solutions and analyzing the impact of vertex weights on the problem's difficulty.

Contribution

It provides complexity analysis and solutions for the edge monitoring problem on multiple graph classes, including generalizations with vertex weights.

Findings

Complexity results for edge monitoring on different graph classes.

Algorithms or methods for specific graph classes.

Impact of vertex weights on the problem's complexity.

Abstract

In this paper, we study the complexity of the edge monitoring problem. A vertex $v$ monitors an edge $e$ if both extremities together with $v$ form a triangle in the graph. Given a graph $G=(V,E)$ and a weight function on edges $c$ where $c(e)$ is the number of monitors that needs the edge $e$, the problem is to seek a minimum subset of monitors $S$ such that every edge $e$ in the graph is monitored by at least $c(e)$ vertices in $S$. In this paper, we study the edge monitoring problem on several graph classes such as complete graphs, block graphs, cographs, split graphs, interval graphs and planar graphs. We also generalize the problem by adding weights on vertices.