License: Creative Commons Attribution 3.0 Unported license (CC-BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SWAT.2018.32
URN: urn:nbn:de:0030-drops-88586
URL: https://drops.dagstuhl.de/opus/volltexte/2018/8858/
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Demaine, Erik D. ; Rudoy, Mikhail

Tree-Residue Vertex-Breaking: a new tool for proving hardness

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LIPIcs-SWAT-2018-32.pdf (0.5 MB)


Abstract

In this paper, we introduce a new problem called Tree-Residue Vertex-Breaking (TRVB): given a multigraph G some of whose vertices are marked "breakable," is it possible to convert G into a tree via a sequence of "vertex-breaking" operations (replacing a degree-k breakable vertex by k degree-1 vertices, disconnecting the k incident edges)? We characterize the computational complexity of TRVB with any combination of the following additional constraints: G must be planar, G must be a simple graph, the degree of every breakable vertex must belong to an allowed list B, and the degree of every unbreakable vertex must belong to an allowed list U. The two results which we expect to be most generally applicable are that (1) TRVB is polynomially solvable when breakable vertices are restricted to have degree at most 3; and (2) for any k >= 4, TRVB is NP-complete when the given multigraph is restricted to be planar and to consist entirely of degree-k breakable vertices. To demonstrate the use of TRVB, we give a simple proof of the known result that Hamiltonicity in max-degree-3 square grid graphs is NP-hard. We also demonstrate a connection between TRVB and the Hypergraph Spanning Tree problem. This connection allows us to show that the Hypergraph Spanning Tree problem in k-uniform 2-regular hypergraphs is NP-complete for any k >= 4, even when the incidence graph of the hypergraph is planar.

BibTeX - Entry

@InProceedings{demaine_et_al:LIPIcs:2018:8858,
  author =	{Erik D. Demaine and Mikhail Rudoy},
  title =	{{Tree-Residue Vertex-Breaking: a new tool for proving hardness}},
  booktitle =	{16th Scandinavian Symposium and Workshops on Algorithm  Theory (SWAT 2018)},
  pages =	{32:1--32:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-068-2},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{101},
  editor =	{David Eppstein},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2018/8858},
  URN =		{urn:nbn:de:0030-drops-88586},
  doi =		{10.4230/LIPIcs.SWAT.2018.32},
  annote =	{Keywords: NP-hardness, graphs, Hamiltonicity, hypergraph spanning tree}
}

Keywords: NP-hardness, graphs, Hamiltonicity, hypergraph spanning tree
Collection: 16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018)
Issue Date: 2018
Date of publication: 04.06.2018


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