Finding large matchings in 1-planar graphs of minimum degree 3

A matching is a set of edges without common endpoint. It was recently shown that every 1-planar graph (i.e., a graph that can be drawn in the plane with at most one crossing per edge) that has minimum degree 3 has a matching of size at least $\frac{n+12}{7}$, and this is tight for some graphs. The proof did not come with an algorithm to find the matching more efficiently than a general-purpose maximum-matching algorithm. In this paper, we give such an algorithm. More generally, we show that any matching that has no augmenting paths of length 9 or less has size at least $\frac{n+12}{7}$ in a 1-planar graph with minimum degree 3

Computing Large Matchings in Planar Graphs with Fixed Minimum Degree

In this paper we present algorithms that compute large matchings in planar graphs with fixed minimum degree. The algorithms give a guarantee on the size of the computed matching and run in linear time. Thus they are faster than the best known algorithm for computing maximum matchings in general graphs and in planar graphs, which run in O ( √ nm) and O(n 1.188) time, respectively. For the class of planar graphs with minimum degree 3 the bounds we achieve are known to be best possible. Further, we discuss how minimum degree 5 can be used to obtain stronger bounds on the matching size