Hydras: Directed Hypergraphs and Horn Formulas
Robert H. Sloan, Despina Stasi, and Gyorgy Turan

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 is the minimal number of hyperarcs of the form required in a directed hypergraph , such that for every pair , the set of vertices reachable in from is the entire vertex set if , and it is 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…
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Taxonomy
TopicsAdvanced Graph Theory Research · Complexity and Algorithms in Graphs · semigroups and automata theory
