The complexity of finding certain trees in tournaments

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 lg^α 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.