Hydras: Complexity on general graphs and a subclass of trees

Hydra formulas were introduced in (Sloan, Stasi, and Turán 2012). A hydra formula is a Horn formula consisting of definite Horn clauses of size 3 specified by a set of bodies of size 2, and containing clauses formed by these bodies and all possible heads. A hydra formula can be specified by the undirected graph formed by the bodies occurring in the formula. The minimal formula size for hydras is then called the hydra number of the underlying graph. In this paper we aim to answer some open questions regarding complexity of determining the hydra number of a graph which were left open in (Sloan, Stasi, and Turán 2012). In particular we show that the problem of checking, whether a graph G = (V,E) is single-headed, i.e. whether the hydra number of G is equal to the number of edges, is NP-complete. We also consider hydra number of trees and we describe a family of trees for which the hydra number can be determined in polynomial time.