Map graphs generalize planar graphs and were introduced by Chen, Grigni and Papadimitriou [STOC 1998, J.ACM 2002]. They showed that the problem of recognizing map graphs is in NP by proving the existence of a planar witness graph W. Shortly after, Thorup [FOCS 1998] published a polynomial-time recognition algorithm for map graphs. However, the run time of this algorithm is estimated to be Omega(n^{120}) for n-vertex graphs, and a full description of its details remains unpublished. We give a new and purely combinatorial algorithm that decides whether a graph G is a map graph having an outerplanar witness W. This is a step towards a first combinatorial recognition algorithm for general map graphs. The algorithm runs in time and space O(n+m). In contrast to Thorup's approach, it computes the witness graph W in the affirmative case.
@InProceedings{mnich_et_al:LIPIcs.SWAT.2016.5, author = {Mnich, Matthias and Rutter, Ignaz and Schmidt, Jens M.}, title = {{Linear-Time Recognition of Map Graphs with Outerplanar Witness}}, booktitle = {15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016)}, pages = {5:1--5:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-011-8}, ISSN = {1868-8969}, year = {2016}, volume = {53}, editor = {Pagh, Rasmus}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2016.5}, URN = {urn:nbn:de:0030-drops-60349}, doi = {10.4230/LIPIcs.SWAT.2016.5}, annote = {Keywords: Algorithms and data structures, map graphs, recognition, planar graphs} }
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