Abstract
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.
F. Klute—This work was conducted while FK was a member of the “Algorithms and Complexity Group” at TU Wien.
Research of TB supported by NSERC. Research initiated while FK was visiting the University of Waterloo.
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Notes
- 1.
In this paper, ‘minimum degree k’ stands for ‘minimum degree at least k’; of course the bounds also hold if all degrees are higher.
- 2.
Our terminology follows the one in Edmonds’ famous blossom-algorithm [12].
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Biedl, T., Klute, F. (2020). Finding Large Matchings in 1-Planar Graphs of Minimum Degree 3. In: Adler, I., Müller, H. (eds) Graph-Theoretic Concepts in Computer Science. WG 2020. Lecture Notes in Computer Science(), vol 12301. Springer, Cham. https://doi.org/10.1007/978-3-030-60440-0_20
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