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Hydras: Directed Hypergraphs and Horn Formulas

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 $u,v\rightarrow w$ 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 $(u, v) \in E$, 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.