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 Π p 2 [HOZZ]. We prove that this bound is optimal: We construct a succinctly specified tournament family whose king problem is Π p 2-complete. It follows easily from our proof approach that the problem of testing kingship in succinctly specified graphs (which need not

The nondeterministic advice complexity of the P-selective sets is known to be exactly linear. Regarding the deterministic advice complexity of the P-selective sets—i.e., the amount of Karp– Lipton advice needed for polynomial-time machines to recognize them in general—the best current upper bound is quadratic [Ko83] and the best current lower bound is linear [HT96]. We prove that every associatively P-selective set is commutatively, associatively P-selective. Using this, we establish an algebraic sufficient condition for the P-selective sets to have a linear upper bound (which thus would match the existing lower bound) on their deterministic advice complexity: If all P-selective sets are associatively P-selective then the deterministic advice complexity of the P-selective sets is linear. The weakest previously known sufficient condition was P = NP. We also establish related results for algebraic properties of, and advice complexity of, the nondeterministically selective sets.

The Hartmanis–Immerman–Sewelson theorem is the classical link between the exponential and the polynomial time realm. It states that NE = E if, and only if, every sparse set in NP lies in P. We establish similar links for classes other than sparse sets: 1. E = UE ⇐ ⇒ all functions f: {1} ∗ → Σ ∗ in NPSVg lie in FP. 2. E = NE ⇐ ⇒ all functions f: {1} ∗ → Σ ∗ in NPFewV lie in FP. 3. E = E NP ⇐ ⇒ all functions f: {1} ∗ → Σ ∗ in OptP lie in FP. 4. E = E NP ⇐ ⇒ all standard left cuts in NP lie in P. 5. E = EH ⇐ ⇒ PH ∩ P/poly = P. We apply these results to the immunity of P-selective sets. It is known that they can be bi-immune, but not Π p 2 /1-immune. Their immunity is closely related to top-Toda languages, whose complexity we link to the exponential realm, and also to king languages. We introduce the new notion of superkings, which are characterized in terms of ∃∀-predicates rather than ∀∃-predicates, and show that king languages cannot be Σ p 2-immune. As a consequence, /1-immune and, if EΣp2 = E, not even P/1-immune. P-selective sets cannot be Σ p 2 1