On the Complexity of Kings

The diameter of an undirected graph is the minimal number d such that there is a path between any two vertices of the graph of length at most d. The radius of a graph is the minimal number r such that there exists a vertex in the graph from which all other vertices can be reached in at most r steps. In the present paper we study the computational complexity of deciding whether a given graph has some fixed small diameter or radius. For graphs given as adjacency matrices the problem is trivial, but for graph that are represented succinctly using circuits, the complexity is more interesting: We show that for every fixed d ≥ 2 the problem of deciding whether a succinctly represented undirected graph has diameter at most d is Πp 2-complete, while for d = 1 the problem is Πp 1-complete; and for every fixed r ≥ 2 the problem of deciding whether a succinctly represented undirected graph has radius at most r is Σp 3-complete, while for r = 1 the problem is Σp 2-complete. This shows ...

• Type: Conference
• Year: 2007
• Where: FCT
• Authors: Edith Hemaspaandra, Lane A. Hemaspaandra, Till Tantau, Osamu Watanabe
• Keywords: Applied Computing, Computational Complexity, FCT 2007, Minimal Number, Undirected Graph