# Efficient algorithms for finding maximum matchings in convex bipartite graphs and related problems

- Received:

DOI: 10.1007/BF00264533

- Cite this article as:
- Lipski, W. & Preparata, F.P. Acta Informatica (1981) 15: 329. doi:10.1007/BF00264533

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## Summary

A bipartite graph *G=(A, B, E)* is convex on the vertex set *A* if *A* can be ordered so that for each element *b* in the vertex set *B* the elements of *A* connected to *b* form an interval of *A*; *G* is doubly convex if it is convex on both *A* and *B.* Letting ¦*A*¦=*m* and ¦*B*¦=*n*, in this paper we describe maximum matching algorithms which run in time *O(m + nA(n))* on convex graphs (where *A(n)* is a very slowly growing function related to a functional inverse of Ackermann's function), and in time *O(m+n)* on doubly convex graphs. We also show that, given a maximum matching in a convex bipartite graph *G*, a corresponding maximum set of independent vertices can be found in time *O(m+n)*. Finally, we briefly discuss some generalizations of convex bipartite graphs and some extensions of the previously discussed techniques to instances in scheduling theory.