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.2019.13
URN: urn:nbn:de:0030-drops-109574
URL: https://drops.dagstuhl.de/opus/volltexte/2019/10957/
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Le, Hoang-Oanh ; Le, Van Bang

Constrained Representations of Map Graphs and Half-Squares

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Abstract

The square of a graph H, denoted H^2, is obtained from H by adding new edges between two distinct vertices whenever their distance in H is two. The half-squares of a bipartite graph B=(X,Y,E_B) are the subgraphs of B^2 induced by the color classes X and Y, B^2[X] and B^2[Y]. For a given graph G=(V,E_G), if G=B^2[V] for some bipartite graph B=(V,W,E_B), then B is a representation of G and W is the set of points in B. If in addition B is planar, then G is also called a map graph and B is a witness of G [Chen, Grigni, Papadimitriou. Map graphs. J. ACM , 49 (2) (2002) 127-138]. While Chen, Grigni, Papadimitriou proved that any map graph G=(V,E_G) has a witness with at most 3|V|-6 points, we show that, given a map graph G and an integer k, deciding if G admits a witness with at most k points is NP-complete. As a by-product, we obtain NP-completeness of edge clique partition on planar graphs; until this present paper, the complexity status of edge clique partition for planar graphs was previously unknown. We also consider half-squares of tree-convex bipartite graphs and prove the following complexity dichotomy: Given a graph G=(V,E_G) and an integer k, deciding if G=B^2[V] for some tree-convex bipartite graph B=(V,W,E_B) with |W|<=k points is NP-complete if G is non-chordal dually chordal and solvable in linear time otherwise. Our proof relies on a characterization of half-squares of tree-convex bipartite graphs, saying that these are precisely the chordal and dually chordal graphs.

BibTeX - Entry

@InProceedings{le_et_al:LIPIcs:2019:10957,
  author =	{Hoang-Oanh Le and Van Bang Le},
  title =	{{Constrained Representations of Map Graphs and Half-Squares}},
  booktitle =	{44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
  pages =	{13:1--13:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-117-7},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{138},
  editor =	{Peter Rossmanith and Pinar Heggernes and Joost-Pieter Katoen},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2019/10957},
  URN =		{urn:nbn:de:0030-drops-109574},
  doi =		{10.4230/LIPIcs.MFCS.2019.13},
  annote =	{Keywords: map graph, half-square, edge clique cover, edge clique partition, graph classes}
}

Keywords: map graph, half-square, edge clique cover, edge clique partition, graph classes
Collection: 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)
Issue Date: 2019
Date of publication: 20.08.2019


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