Hardness of Finding Kings and Strong Kings

Abstract

A king in a directed graph is a vertex $v$ such that every other vertex is reachable from $v$ via a path of length at most $2$. It is well known that every tournament (a complete graph where each edge has a direction) has at least one king. Our contributions in this work are: - We show that the query complexity of determining existence of a king in arbitrary $n$-vertex digraphs is $\Theta(n^2)$. This is in stark contrast to the case where the input is a tournament, where Shen, Sheng, and Wu [SICOMP'03] showed that a king can be found in $O(n^{3/2})$ queries. - In an attempt to increase the "fairness" in the definition of tournament winners, Ho and Chang [IPL'03] defined a strong king to be a king $k$ such that, for every $v$ that dominates $k$, the number of length-$2$ paths from $k$ to $v$ is strictly larger than the number of length-$2$ paths from $v$ to $k$. We show that the query complexity of finding a strong king in a tournament is $\Theta(n^2)$. This answers a question of Biswas, Jayapaul, Raman, and Satti [DAM'22] in the negative. A key component in our proofs is the design of specific tournaments where every vertex is a king, and analyzing certain properties of these tournaments. We feel these constructions and properties are independently interesting and may lead to more interesting results about tournament solutions.