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 $k \geq 2$

On the existence and number of (k+1)(k+1)-kings in kk-quasi-transitive digraphs

Let $D=(V(D), A(D))$ be a digraph and $k \ge 2$ an integer. We say that $D$ is $k$-quasi-transitive if for every directed path $(v_0, v_1,..., v_k)$ in $D$, then $(v_0, v_k) \in A(D)$ or $(v_k, v_0) \in A(D)$. Clearly, a 2-quasi-transitive digraph is a quasi-transitive digraph in the usual sense. Bang-Jensen and Gutin proved that a quasi-transitive digraph $D$ has a 3-king if and only if $D$ has a unique initial strong component and, if $D$ has a 3-king and the unique initial strong component of $D$ has at least three vertices, then $D$ has at least three 3-kings. In this paper we prove the following generalization: A $k$-quasi-transitive digraph $D$ has a $(k+1)$-king if and only if $D$ has a unique initial strong component, and if $D$ has a $(k+1)$-king then, either all the vertices of the unique initial strong components are $(k+1)$-kings or the number of $(k+1)$-kings in $D$ is at least $(k+2)$.Comment: 17 page

Parameterized Algorithms for Directed Maximum Leaf Problems

We prove that finding a rooted subtree with at least $k$ leaves in a digraph is a fixed parameter tractable problem. A similar result holds for finding rooted spanning trees with many leaves in digraphs from a wide family $\cal L$ that includes all strong and acyclic digraphs. This settles completely an open question of Fellows and solves another one for digraphs in $\cal L$. Our algorithms are based on the following combinatorial result which can be viewed as a generalization of many results for a `spanning tree with many leaves' in the undirected case, and which is interesting on its own: If a digraph $D\in \cal L$ of order $n$ with minimum in-degree at least 3 contains a rooted spanning tree, then $D$ contains one with at least $(n/2)^{1/5}-1$ leaves

A new perspective from hypertournaments to tournaments

A $k$-tournament $H$ on $n$ vertices is a pair $(V, A)$ for $2\leq k\leq n$, where $V(H)$ is a set of vertices, and $A(H)$ is a set of all possible $k$-tuples of vertices, such that for any $k$-subset $S$ of $V$, $A(H)$ contains exactly one of the $k!$ possible permutations of $S$. In this paper, we investigate the relationship between a hyperdigraph and its corresponding normal digraph. Particularly, drawing on a result from Gutin and Yeo, we establish an intrinsic relationship between a strong $k$-tournament and a strong tournament, which enables us to provide an alternative (more straightforward and concise) proof for some previously known results and get some new results.Comment: 10 page