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.
 Kathie Cameron. Induced matchings. Discrete Applied Mathematics,
 Paul C Gilmore and Alan J Hoffman. A characterization of
comparability graphs and of interval graphs. Canad. J. Math,
 Martin Charles Golumbic. Algorithmic graph theory and perfect graphs,
volume 57. Elsevier, 2004.
 Martin Charles Golumbic, Tirza Hirst, and Moshe Lewenstein. Uniquely
restricted matchings. Algorithmica, 31(2):139–154, 2001.
 Martin Charles Golumbic and Renu C Laskar. Irredundancy in circular
arc graphs. Discrete Applied Mathematics, 44(1):79–89, 1993.
 Martin Charles Golumbic and Moshe Lewenstein. New results on
induced matchings. Discrete Applied Mathematics, 101(1):157–165,
 Daniel Hershkowitz and Hans Schneider. Ranks of zero patterns and
sign patterns. Linear and Multilinear Algebra, 34(1):3–19, 1993.
 Vadim E Levit and Eugen Mandrescu. Unicycle graphs and
uniquely restricted maximum matchings. Electronic Notes in Discrete
Mathematics, 22:261–265, 2005.
 Vadim E Levit and Eugen Mandrescu. On unicyclic graphs with
uniquely restricted maximum matchings. Graphs and Combinatorics,
 L´aszl´o Lov´asz and Michael D Plummer. Matching theory, volume 367.
American Mathematical Soc., 2009.
 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.
 Guohun Zhu. Determining all maximum uniquely restricted matching
in bipartite graphs. arXiv preprint arXiv:1009.5435, 2010.