The Complexity of Kings
<|reference_start|>The Complexity of Kings: A king in a directed graph is a node from which each node in the graph can be reached via paths of length at most two. There is a broad literature on tournaments (completely oriented digraphs), and it has been known for more than half a century that all tournaments have at least one king [Lan53]. Recently, kings have proven useful in theoretical computer science, in particular in the study of the complexity of the semifeasible sets [HNP98,HT05] and in the study of the complexity of reachability problems [Tan01,NT02].   In this paper, we study the complexity of recognizing kings. For each succinctly specified family of tournaments, the king problem is known to belong to $\Pi_2^p$ [HOZZ]. We prove that this bound is optimal: We construct a succinctly specified tournament family whose king problem is $\Pi_2^p$-complete. It follows easily from our proof approach that the problem of testing kingship in succinctly specified graphs (which need not be tournaments) is $\Pi_2^p$-complete. We also obtain $\Pi_2^p$-completeness results for k-kings in succinctly specified j-partite tournaments, $k,j \geq 2$, and we generalize our main construction to show that $\Pi_2^p$-completeness holds for testing k-kingship in succinctly specified families of tournaments for all $k \geq 2$.<|reference_end|>
@article{hemaspaandra2005the,
    title={The Complexity of Kings},
    author={Edith Hemaspaandra, Lane A. Hemaspaandra, Osamu Watanabe},
    journal={arXiv preprint arXiv:cs/0506055},
    year={2005},
    number={URCS-TR-2005-870},
    archivePrefix={arXiv},
    eprint={cs/0506055},
    primaryClass={cs.CC cs.DM}
}