Abstract
A tournament T n is an orientation of the complete graph on n vertices. We continue the algorithmic study initiated by Hell and Rosenfeld[5] of recognizing various directed trees in tournaments. Hell and Rosenfeld considered orientations of paths, and showed the existence of oriented paths on n vertices finding which in T n requires Θ(n lgα n) “edge probes” where α ≤ 1 is any fixed non-negative constant. Here, we investigate the complexity of finding a vertex of prescribed outdegree (or indegree). In particular, we show, by proving upper and lower bounds, that the complexity of finding a vertex of outdegree k(n−1)/2) in T n is Θ(nk). We also establish an Ω(n 2) lower bound for finding a vertex of maximum outdegree in T n. These bounds are in sharp contrast to the O(n) bounds for selection in the case of transitive tournaments.
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