Disjoint paths in unions of tournaments

TL;DR

This paper extends polynomial-time algorithms for vertex-disjoint paths problems from tournaments to more general digraphs partitioned into semicomplete subgraphs, addressing a complex routing problem.

Contribution

It introduces a polynomial-time algorithm for vertex-disjoint paths in digraphs partitioned into a bounded number of semicomplete subgraphs, generalizing previous results.

Findings

Polynomial-time algorithm for fixed k in partitioned digraphs

Extension of disjoint paths problem to broader classes of digraphs

Addresses a complex routing problem in directed graphs

Abstract

Given $k$ pairs of vertices $(s_i,t_i)\;(1\le i\le k)$ of a digraph $G$, how can we test whether there exist vertex-disjoint directed paths from $s_i$ to $t_i$ for $1\le i\le k$? This is NP-complete in general digraphs, even for $k = 2$, but in an earlier paper we proved that for all fixed $k$, there is a polynomial-time algorithm to solve the problem if $G$ is a tournament (or more generally, a semicomplete digraph). Here we prove that for all fixed $k$ there is a polynomial-time algorithm to solve the problem when $V(G)$ is partitioned into a bounded number of sets each inducing a semicomplete digraph (and we are given the partition).