Intersection Graphs of Pseudosegments: Chordal Graphs
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Publication:3075604
DOI10.7155/JGAA.00204zbMATH Open1213.05106arXiv0809.1980OpenAlexW2963942884MaRDI QIDQ3075604
Stefan Felsner, Cornelia Dangelmayr, William T. jun. Trotter
Publication date: 16 February 2011
Published in: Journal of Graph Algorithms and Applications (Search for Journal in Brave)
Abstract: We investigate which chordal graphs have a representation as intersection graphs of pseudosegments. For positive we have a construction which shows that all chordal graphs that can be represented as intersection graph of subpaths on a tree are pseudosegment intersection graphs. We then study the limits of representability. We describe a family of intersection graphs of substars of a star which is not representable as intersection graph of pseudosegments. The degree of the substars in this example, however, has to get large. A more intricate analysis involving a Ramsey argument shows that even in the class of intersection graphs of substars of degree three of a star there are graphs that are not representable as intersection graph of pseudosegments. Motivated by representability questions for chordal graphs we consider how many combinatorially different k-segments, i.e., curves crossing k distinct lines, an arrangement of n pseudolines can host. We show that for fixed k this number is in O(n^2). This result is based on a k-zone theorem for arrangements of pseudolines that should be of independent interest.
Full work available at URL: https://arxiv.org/abs/0809.1980
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