Computing Maximum Uniquely Restricted Matchings in Restricted Interval Graphs

Authors: Swapnil Gupta, C. Pandu Rangan

Abstract:

A uniquely restricted matching is defined to be a matching M whose matched vertices induces a sub-graph which has only one perfect matching. In this paper, we make progress on the open question of the status of this problem on interval graphs (graphs obtained as the intersection graph of intervals on a line). We give an algorithm to compute maximum cardinality uniquely restricted matchings on certain sub-classes of interval graphs. We consider two sub-classes of interval graphs, the former contained in the latter, and give O(|E|^2) time algorithms for both of them. It is to be noted that both sub-classes are incomparable to proper interval graphs (graphs obtained as the intersection graph of intervals in which no interval completely contains another interval), on which the problem can be solved in polynomial time.

Keywords: Uniquely restricted matching, interval graph, design and analysis of algorithms, matching, induced matching, witness counting.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1124469

References:

[1] Kathie Cameron. Induced matchings. Discrete Applied Mathematics, 24(1):97–102, 1989.
[2] Paul C Gilmore and Alan J Hoffman. A characterization of comparability graphs and of interval graphs. Canad. J. Math, 16(539-548):4, 1964.
[3] Martin Charles Golumbic. Algorithmic graph theory and perfect graphs, volume 57. Elsevier, 2004.
[4] Martin Charles Golumbic, Tirza Hirst, and Moshe Lewenstein. Uniquely restricted matchings. Algorithmica, 31(2):139–154, 2001.
[5] Martin Charles Golumbic and Renu C Laskar. Irredundancy in circular arc graphs. Discrete Applied Mathematics, 44(1):79–89, 1993.
[6] Martin Charles Golumbic and Moshe Lewenstein. New results on induced matchings. Discrete Applied Mathematics, 101(1):157–165, 2000.
[7] Daniel Hershkowitz and Hans Schneider. Ranks of zero patterns and sign patterns. Linear and Multilinear Algebra, 34(1):3–19, 1993.
[8] Vadim E Levit and Eugen Mandrescu. Unicycle graphs and uniquely restricted maximum matchings. Electronic Notes in Discrete Mathematics, 22:261–265, 2005.
[9] Vadim E Levit and Eugen Mandrescu. On unicyclic graphs with uniquely restricted maximum matchings. Graphs and Combinatorics, 29(6):1867–1879, 2013.
[10] L´aszl´o Lov´asz and Michael D Plummer. Matching theory, volume 367. American Mathematical Soc., 2009.
[11] Lucia Draque Penso, Dieter Rautenbach, and Ueverton dos Santos Souza. Graphs in which some and every maximum matching is uniquely restricted. arXiv preprint arXiv:1504.02250, 2015.
[12] Guohun Zhu. Determining all maximum uniquely restricted matching in bipartite graphs. arXiv preprint arXiv:1009.5435, 2010.