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When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.MFCS.2019.13
URN: urn:nbn:de:0030-drops-109574
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2019/10957/
Le, Hoang-Oanh ;
Le, Van Bang
Constrained Representations of Map Graphs and Half-Squares
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}
}
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Keywords: |
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map graph, half-square, edge clique cover, edge clique partition, graph classes |
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Collection: |
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44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019) |
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Issue Date: |
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2019 |
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Date of publication: |
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20.08.2019 |
