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Algorithmica

, Volume 31, Issue 2, pp 139–154 | Cite as

Uniquely Restricted Matchings

  • M. C. Golumbic
  • T. Hirst
  • M. Lewenstein
Article

Abstract.

A matching in a graph is a set of edges no two of which share a common vertex. In this paper we introduce a new, specialized type of matching which we call uniquely restricted matchings, originally motivated by the problem of determining a lower bound on the rank of a matrix having a specified zero/ non-zero pattern. A uniquely restricted matching is defined to be a matching M whose saturated vertices induce a subgraph which has only one perfect matching, namely M itself. We introduce the two problems of recognizing a uniquely restricted matching and of finding a maximum uniquely restricted matching in a given graph, and present algorithms and complexity results for certain special classes of graphs. We demonstrate that testing whether a given matching M is uniquely restricted can be done in O(|M||E|) time for an arbitrary graph G=(V,E) and in linear time for cacti, interval graphs, bipartite graphs, split graphs and threshold graphs. The maximum uniquely restricted matching problem is shown to be NP-complete for bipartite graphs, split graphs, and hence for chordal graphs and comparability graphs, but can be solved in linear time for threshold graphs, proper interval graphs, cacti and block graphs.

Key words. Graph theory, Perfect graphs, Design and analysis of algorithms, Matchings, Uniquely restricted matchings. 

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Copyright information

© Springer-Verlag New York Inc. 2001

Authors and Affiliations

  • M. C. Golumbic
    • 1
  • T. Hirst
    • 1
  • M. Lewenstein
    • 2
  1. 1.Department of Mathematics and Computer Science, Bar-Ilan University, 52900 Ramat-Gan, Israel. \{golumbic,tirza\}@cs.biu.ac.il.IL
  2. 2.IBM Research, P.O. Box 218, Yorktown Heights, NY 10598, USA. moshe@watson.ibm.com.US

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