Efficient maximum matching algorithms for trapezoid graphs

Phan-Thuan Do (Department of Computer Science, Hanoi University of Science and Technology, Vietnam)
Ngoc-Khang Le (LIP, ENS de Lyon, Lyon, France)
Van-Thieu Vu (Department of Computer Science, Hanoi University of Science and Technology, Vietnam)

Electronic Journal of Graph Theory and Applications (EJGTA)
Vol 5, No 1 (2017): Electronic Journal of Graph Theory and Applications

Trapezoid graphs are intersection graphs of trapezoids between two horizontal lines. Many NP-hard problems can be solved in polynomial time if they are restricted on trapezoid graphs. A matching in a graph is a set of pairwise disjoint edges, and a maximum matching is a matching of maximum size. In this paper, we first propose an $O(n(\log n)^3)$ algorithm for finding a maximum matching in trapezoid graphs, then improve the complexity to $O(n(\log n)^2)$. Finally, we generalize this algorithm to a larger graph class, namely $k$-trapezoid graphs. To the best of our knowledge, these are the first efficient maximum matching algorithms for trapezoid graphs.