Monitoring arc-geodetic sets of oriented graphs

2024-08-31

Theoretical Computer Science

AI Summary

This paper introduces the novel concept of a *monitoring arc-geodetic set*—a minimum vertex subset in a directed graph that uniquely identifies every arc via its geodesic paths. Methodologically, it integrates combinatorial graph theory with directed geodesy to derive exact monitoring numbers for several graph classes, including tournaments, DAGs, trees, and bipartite graphs. It establishes NP-completeness for deciding the existence of an arc-geodetic monitoring set of size at most $k$ in directed graphs with maximum degree at most 4, and proves APX-hardness of the general optimization problem. Additionally, it devises an $O(log n)$-approximation algorithm. The work provides a tight combinatorial characterization linking directed graph structure to monitoring coverage, thereby establishing a foundational framework and algorithmic toolkit for extending edge-monitoring theory to the directed setting.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Topic discovery and tracking

Problem

• Extend monitoring sets to oriented graphs.
• Define and compute arc-geodetic numbers.
• Analyze complexity of related algorithmic problems.

Innovation

• Extends monitoring to oriented graphs
• Defines arc-geodetic numbers for graphs
• Proves NP-hardness for specific orientations