Reachability and Matching in Single Crossing Minor Free Graphs

TL;DR

This paper demonstrates that for graphs excluding a single crossing minor, key problems like reachability and maximum matching can be efficiently solved in small complexity classes, extending previous results.

Contribution

It introduces a polynomially bounded weight function for all $H$-minor free graphs with a single crossing $H$, enabling efficient algorithms for reachability and matching.

Findings

Reachability in UL for $H$-minor free graphs

Maximum bipartite matching in SPL for these graphs

Improves NC bounds for bipartite perfect matching in this class

Abstract

We show that for each single crossing graph $H$, a polynomially bounded weight function for all $H$-minor free graphs $G$ can be constructed in Logspace such that it gives nonzero weights to all the cycles in $G$. This class of graphs subsumes almost all classes of graphs for which such a weight function is known to be constructed in Logspace. As a consequence, we obtain that for the class of $H$-minor free graphs where $H$ is a single crossing graph, reachability can be solved in UL, and bipartite maximum matching can be solved in SPL, which are small subclasses of the parallel complexity class NC. In the restrictive case of bipartite graphs, our maximum matching result improves upon the recent result of Eppstein and Vazirani, where they show an NC bound for constructing perfect matching in general single crossing minor free graphs.