# NC Algorithms for Computing a Perfect Matching and a Maximum Flow in One-Crossing-Minor-Free Graphs

@article{Eppstein2018NCAF,
title={NC Algorithms for Computing a Perfect Matching and a Maximum Flow in One-Crossing-Minor-Free Graphs},
author={David Eppstein and Vijay V. Vazirani},
journal={SIAM J. Comput.},
year={2018},
volume={50},
pages={1014-1033}
}
• Published 31 January 2018
• Mathematics
• SIAM J. Comput.
In 1988, Vazirani gave an NC algorithm for computing the number of perfect matchings in $K_{3,3}$-minor-free graphs by building on Kasteleyn's scheme for planar graphs, and stated that this "opens up the possibility of obtaining an NC algorithm for finding a perfect matching in $K_{3,3}$-free graphs." In this paper, we finally settle this 30-year-old open problem. Building on recent NC algorithms for planar and bounded-genus perfect matching by Anari and Vazirani and later by Sankowski, we…
2 Citations

## Figures from this paper

• Mathematics
SODA
• 2019
This work considers decomposing a 3-connected planar graph G using laminar separators of size three and shows how to find a maximal set of 3-separators disjoint from v which are laminAR and satisfy: every vertex in a separator $X$ has two neighbours not in the unique component of G-X containing $v$.
• Mathematics
FSTTCS
• 2021
The 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.

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