On the complexity of kings A king in a directed graph is a vertex from which each vertex in the graph can be reached through 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. Recently, kings have proven useful in theoretical computer science, in particular in the study of the complexity of reachability problems and semifeasible sets. In this article, we study the complexity of recognizing kings. For each succinctly specified family of tournaments, the king problem is already known to belong to P2p. We prove that the complexity of kingship problems is a rich enough vocabulary to pinpoint every nontrivial manyone degree in P2p. That is, we show that every set in P2p other than 0? and S* is equivalent to a king problem under ?mpreductions. Indeed, we show that the equivalence can even be realized by relatively simple padding, and holds even if the notion of kings is redefined to refer to kkings (for any fixed ku003e2)vertices from which all vertices can be reached through paths of length at most k. In contrast, we prove that for each succinctly specified family of tournaments the source problem (the problem of deciding whether a given vertex v has the property that there exists a k such that v is a kking) also falls within P2p, yet cannot be P2pcompleteor even NPhardunless PNP. Using these and related techniques, we obtain a broad range of additional results about the complexity of king problems, diameter problems, and radius problems. It follows easily from our proof approach that the problem of testing kingship in succinctly specified graphs (which need not be tournaments) is P2pcomplete. We show that the radius problem for arbitrary succinctly represented graphs is S3pcomplete, but that in contrast the diameter problem for arbitrary succinctly represented graphs (or even tournaments) is P2pcomplete.

On the Complexity of Digraph Colourings and Vertex Arboricity It has been shown by Bokal et al. that deciding 2colourability of  :12,an NPcomplete problem. This result was later on extended by Feder et al.  that deciding :26,whether a digraph has a circular pcolouring  :36,91,110,131,for all rational pu003e1. In this paper, we consider the  :47,corresponding decision problems :36,91,110,131,for related notions of fractional  :56,digraphs and graphs, including the star dichromatic number, the  :66,number and the circular vertex arboricity. We prove the   :76,Deciding if the star dichromatic :66,number of a digraph is at most p  :36,91,110,131,for every rational   :76,Deciding if the fractional dichromatic :66,number of a digraph is at most p  :36,91,110,131,for every :95,135,pu003e1, p neq   :76,Deciding if the circular vertex :116,arboricity of a graph is at most p  :36,91,110,131,for every rational   To show these results, different techniques are required in each case.  :148,to prove the first result, we relate the star dichromatic :66,number :148,to a  of :162,homomorphisms between digraphs, called circular homomorphisms,  :170,be of independent interest. We provide a classification of  :180,complexities of the :47,corresponding homomorphism colouring  :148,to the :187,one derived by Feder et al. :36,91,110,131,for acyclic homomorphisms.