Computing large matchings fast

Abstract

In this paper we present algorithms for computing large matchings in 3-regular graphs, graphs with maximum degree 3, and 3-connected planar graphs. The algorithms give a guarantee on the size of the computed matching and take linear or slightly superlinear time. Thus they are faster than the best-known algorithm for computing maximum matchings in general graphs, which runs in O( √ nm) time, where n denotes the number of vertices and m the number of edges of the given graph. For the classes of 3-regular graphs and graphs with maximum degree 3 the bounds we achieve are known to be best possible.We also investigate graphs with block trees of bounded degree, where the d-block tree is the adjacency graph of the d-connected components of the given graph. In 3-regular graphs and 3-connected planar graphs with bounded-degree 2-and 4-block trees, respectively, we show how to compute maximum matchings in slightly superlinear time.

Computing large matchings in planar graphs with fixed minimum degree

“…In fact, the question how fixed minimum degrees can be exploited algorithmically was posed as an open question in [24]. We answer this question and show that the tight bounds of Nishizeki and Baybars [22] for minimum degree 3 can be reached in linear time.…”

“…Recently, Rutter and Wolff [24] (a preliminary version appeared in [25]) gave fast algorithms that achieve the tight bounds of Biedl et al Their algorithms compute matchings of size (n − 1)/3 in maxdeg-3 graphs in linear time, of size (4n − 1)/9 in 3-regular graphs in O(n log 4 n) time, and of size (n + 4)/3 in 3-connected planar graphs in linear time. For graphs with bounded maximum degree k, lower bounds for the size of maximal matchings have been considered [26].…”

Computing Large Matchings in Planar Graphs with Fixed Minimum Degree

“…In fact the question how fixed minimum degrees can be exploited algorithmically was posed as an open question in [21]. We answer this question and show that the tight bounds of Nishizeki and Baybars [17] for minimum degree 3 can be reached in linear time.…”

“…Recently, Rutter and Wolff [21] (a preliminary version appeared in [20]) gave fast algorithms that achieve the tight bounds of Biedl et al Their algorithms compute matchings of size (n − 1)/3 in maxdeg-3 graphs in linear time, of size (4n − 1)/9 in 3-regular graphs in O(n log 4 n) time and of size (n + 4)/3 in 3-connected planar graphs in linear time. For graphs with bounded maximum degree k lower bounds for the size of maximal matchings where considered [9].…”