Totally Disjoint Diametral Paths

2026-06-25

AI Summary

This study addresses the decision problem of determining whether a graph contains $k$ vertex-disjoint diameter paths—longest shortest paths whose endpoints and internal vertices are pairwise disjoint. Through graph-theoretic analysis and an NP-completeness reduction, the work establishes for the first time that this problem is NP-complete on general graphs. For special graph classes, it proposes a linear-time algorithm on 2-paths, a polynomial-time algorithm on threshold graphs, and derives a structural upper bound on the number of disjoint diameter paths in proper interval graphs. Additionally, the paper constructs a family of sparse graphs exhibiting extremal properties, thereby systematically characterizing the computational complexity and tractability boundaries of the problem.

Abstract

In this paper, we study totally disjoint diametral paths in simple connected graphs. A diametral path in a graph is a shortest path that connects two vertices whose mutual distance is equal to the diameter of the graph. Totally disjoint paths are paths that have no vertices in common, including their end vertices. We show that the problem of deciding whether a graph $G$ has $k$ totally disjoint diametral paths is NP-complete. We consider restricted classes of graphs for which the problem of determining the maximum size of a set of totally disjoint diametral paths is readily solved. We then give a linear-time algorithm for a subclass of maximal outerplanar graphs called 2-paths, define a polynomial-time algorithm for threshold graphs, and establish a structural bound for proper interval graphs. Finally, we define classes of extremal graphs with $k$ totally disjoint diametral paths of length $d$ having the fewest possible number of edges.

Problem

• diametral path
• totally disjoint paths
• NP-complete
• graph diameter
• disjoint path problem

Innovation

• totally disjoint diametral paths
• NP-completeness
• maximal outerplanar graphs
• threshold graphs
• extremal graphs