Hydras: Directed hypergraphs and Horn formulas

Abstract

We introduce a new graph parameter, the hydra number, arising from the minimization problem for Horn formulas in propositional logic. The hydra number of a graph G=(V,E) is the minimal number of hyperarcs of the form u,v→w required in a directed hypergraph H=(V,F), such that for every pair (u,v), the set of vertices reachable in H from {u,v} is the entire vertex set V if (u,v)∈E, and it is {u,v} otherwise. Here reachability is defined by forward chaining, a standard marking algorithm.Various bounds are given for the hydra number. We show that the hydra number of a graph can be upper bounded by the number of edges plus the path cover number of the line graph of a spanning subgraph, which is a sharp bound in several cases. On the other hand, we construct single-headed graphs for which that bound is off by a constant factor. Furthermore, we characterize trees with low hydra number, and give a lower bound for the hydra number of trees based on the number of vertices that are leaves in the tree obtained from T by deleting its leaves. This bound is sharp for some families of trees. We give bounds for the hydra number of complete binary trees and also discuss a related minimization problem.