When you come at the kings you best not miss
Oded Lachish, Felix Reidl, Chhaya Trehan

TL;DR
This paper introduces a new strategy for identifying a nearly-king vertex in a tournament with fewer queries, improving the known bounds by leveraging a novel structural result.
Contribution
It presents a strategy that finds a -king using O(n^{4/3} polylog n) queries, improving previous bounds and introducing a new structural theorem for tournaments.
Findings
Achieves -king identification with fewer queries.
Provides a new structural result for tournaments.
Improves upon previous query complexity bounds.
Abstract
A tournament is an orientation of a complete graph. We say that a vertex in a tournament controls another vertex if there exists a directed path of length at most two from to . A vertex is called a king if it controls every vertex of the tournament. It is well known that every tournament has a king. We follow Shen, Sheng, and Wu (SIAM J. Comput., 2003) in investigating the query complexity of finding a king, that is, the number of arcs in one has to know in order to surely identify at least one vertex as a king. The aforementioned authors showed that one always has to query at least arcs and provided a strategy that queries at most . While this upper bound has not yet been improved for the original problem, Biswas et al. (Frontiers in Algorithmics, 2017) proved that with queries one can identify a…
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Taxonomy
TopicsAdvanced Graph Theory Research · Complexity and Algorithms in Graphs · Graph Theory and Algorithms
