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Hydras: Directed hypergraphs and Horn formulas.

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Title: Hydras: Directed hypergraphs and Horn formulas.
Authors: Sloan, Robert H.1 sloan@uic.edu, Stasi, Despina2 stasdes@iit.edu, Turán, György1,3 gyt@uic.edu
Source: Theoretical Computer Science. Jan2017 Part B, Vol. 658, p417-428. 12p.
Subjects: Hypergraphs, Fuzzy hypergraphs, Graph theory, Bipartite graphs, Number theory
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. [ABSTRACT FROM AUTHOR]
Copyright of Theoretical Computer Science is the property of Elsevier B.V. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: Hydras: Directed hypergraphs and Horn formulas.
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  Data: <searchLink fieldCode="AR" term="%22Sloan%2C+Robert+H%2E%22">Sloan, Robert H.</searchLink><relatesTo>1</relatesTo><i> sloan@uic.edu</i><br /><searchLink fieldCode="AR" term="%22Stasi%2C+Despina%22">Stasi, Despina</searchLink><relatesTo>2</relatesTo><i> stasdes@iit.edu</i><br /><searchLink fieldCode="AR" term="%22Turán%2C+György%22">Turán, György</searchLink><relatesTo>1,3</relatesTo><i> gyt@uic.edu</i>
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  Data: <searchLink fieldCode="JN" term="%22Theoretical+Computer+Science%22">Theoretical Computer Science</searchLink>. Jan2017 Part B, Vol. 658, p417-428. 12p.
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  Data: <searchLink fieldCode="DE" term="%22Hypergraphs%22">Hypergraphs</searchLink><br /><searchLink fieldCode="DE" term="%22Fuzzy+hypergraphs%22">Fuzzy hypergraphs</searchLink><br /><searchLink fieldCode="DE" term="%22Graph+theory%22">Graph theory</searchLink><br /><searchLink fieldCode="DE" term="%22Bipartite+graphs%22">Bipartite graphs</searchLink><br /><searchLink fieldCode="DE" term="%22Number+theory%22">Number theory</searchLink>
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  Data: 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. [ABSTRACT FROM AUTHOR]
– Name: AbstractSuppliedCopyright
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  Data: <i>Copyright of Theoretical Computer Science is the property of Elsevier B.V. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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        Value: 10.1016/j.tcs.2016.05.036
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      – Code: eng
        Text: English
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        PageCount: 12
        StartPage: 417
    Subjects:
      – SubjectFull: Hypergraphs
        Type: general
      – SubjectFull: Fuzzy hypergraphs
        Type: general
      – SubjectFull: Graph theory
        Type: general
      – SubjectFull: Bipartite graphs
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      – SubjectFull: Number theory
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      – TitleFull: Hydras: Directed hypergraphs and Horn formulas.
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              M: 01
              Text: Jan2017 Part B
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              Y: 2017
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