On the Matching Problem for Special Graph Classes

Abstract

An even cycle in a graph is called {\em nice} by Lov{\'a}sz and Plummer in [LP86] if the graph obtained by deleting all vertices of the cycle has some perfect matching. In the present paper we prove some new complexity bounds for various versions of problems related to perfect matchings in graphs with a polynomially bounded number of nice cycles. We show that for graphs with a polynomially bounded number of nice cycles the perfect matching decision problem is in $SPL$, it is hard for $FewL$, and the perfect matching construction problem is in $L^{C_=L} \cap \oplus L$. Furthermore, we significantly improve the best known upper bounds, proved by Agrawal, Hoang, and Thierauf in the STACS'07-paper [AHT07], for the polynomially bounded perfect matching problem by showing that the construction and the counting versions are in $C_=L \cap \oplus L$ and in $C_=L$, respectively. Note that $SPL, \oplus L, C_=L, $ and $L^{C_=L}$ are contained in $NC^2$. Moreover, we show that the problem of computing a maximum matching for bipartite planar graphs is in $L^{C_=L}$. This solves Open Question 4.7 stated in the STACS'08-paper by Datta, Kulkarni, and Roy [DKR08] where it is asked whether computing a maximum matching even for bipartite planar graphs can be done in $NC$. We also show that the problem of computing a maximum matching for graphs with a polynomially bounded number of even cycles is in $L^{C_=L}$.