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In this paper, we study the maximum matching problem in RDV graphs, i. e. , graphs that are vertex-intersection graphs of downward paths in a rooted tree. We show that this problem can be reduced to a problem of testing (repeatedly) whether a vertical segment intersects one of a dynamically changing set of horizontal segments, which in turn reduces to an orthogonal ray shooting query. Using a suitable data structure, we can therefore find a maximum matching in O (n n) time (presuming a linear-sized representation of the graph is given), i. e. , without even looking at all edges.
Biedl et al. (Wed,) studied this question.