Hydras: Directed Hypergraphs and Horn Formulas

TL;DR

This paper introduces the hydra number, a new graph parameter related to Horn formulas, and explores bounds, characterizations, and specific cases such as trees and binary trees, advancing understanding of directed hypergraph representations.

Contribution

It defines the hydra number, establishes bounds, characterizes trees with low hydra number, and analyzes specific graph classes, providing new insights into hypergraph representations of graphs.

Findings

Hydra number can be bounded by edges plus line graph path cover number.

Constructed graphs show bounds can be off by a constant factor.

Characterized trees with low hydra number and provided bounds for binary trees.

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 $u,v\rightarrow w$ 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.