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.MFCS.2017.28
URN: urn:nbn:de:0030-drops-81289
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2017/8128/
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Milanic, Martin ; Mursic, Peter ; Mydlarz, Marcelo

Induced Embeddings into Hamming Graphs

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LIPIcs-MFCS-2017-28.pdf (0.5 MB)

Abstract

Let d be a positive integer. Can a given graph G be realized in R^d so that vertices are mapped to distinct points, two vertices being adjacent if and only if the corresponding points lie on a common line that is parallel to some axis? Graphs admitting such realizations have been studied in the literature for decades under different names. Peterson asked in [Discrete Appl. Math., 2003] about the complexity of the recognition problem. While the two-dimensional case corresponds to the class of line graphs of bipartite graphs and is well-understood, the complexity question has remained open for all higher dimensions.

In this paper, we answer this question. We establish the NP-completeness of the recognition problem for any fixed dimension, even in the class of bipartite graphs. To do this, we strengthen a characterization of induced subgraphs of 3-dimensional Hamming graphs due to Klavžar and Peterin. We complement the hardness result by showing that for some important classes of perfect graphs –including chordal graphs and distance-hereditary graphs– the minimum dimension of the Euclidean space in which the graph can be realized, or the impossibility of doing so, can be determined in linear time.

BibTeX - Entry

@InProceedings{milanic_et_al:LIPIcs:2017:8128,
  author =	{Martin Milanic and Peter Mursic and Marcelo Mydlarz},
  title =	{{Induced Embeddings into Hamming Graphs}},
  booktitle =	{42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)},
  pages =	{28:1--28:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-046-0},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{83},
  editor =	{Kim G. Larsen and Hans L. Bodlaender and Jean-Francois Raskin},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2017/8128},
  URN =		{urn:nbn:de:0030-drops-81289},
  doi =		{10.4230/LIPIcs.MFCS.2017.28},
  annote =	{Keywords: gridline graph, Hamming graph, induced embedding, NP-completeness, chordal graph}
}

Keywords: gridline graph, Hamming graph, induced embedding, NP-completeness, chordal graph
Collection: 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)
Issue Date: 2017
Date of publication: 01.12.2017

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