# A Linear-Time Algorithm for Maximum-Cardinality Matching on Cocomparability Graphs

## TL;DR

This paper presents a linear-time algorithm for finding maximum-cardinality matchings specifically in cocomparability graphs, a significant subclass of perfect graphs, utilizing the recently developed Lexicographic Depth First Search (LDFS).

## Contribution

It introduces the first linear-time algorithm for maximum-cardinality matching on cocomparability graphs, improving over the general $O(m \sqrt{n})$ solutions.

## Key findings

- Algorithm runs in linear time for cocomparability graphs.
- Utilizes Lexicographic Depth First Search (LDFS) technique.
- Advances matching algorithms for a broad class of perfect graphs.

## Abstract

Finding maximum-cardinality matchings in undirected graphs is arguably one of the most central graph problems. For general m-edge and n-vertex graphs, it is well-known to be solvable in $O(m \sqrt{n})$ time. We develop a linear-time algorithm to find maximum-cardinality matchings on cocomparability graphs, a prominent subclass of perfect graphs that contains interval graphs as well as permutation graphs. Our algorithm is based on the recently discovered Lexicographic Depth First Search (LDFS).