Conference Paper
Triangle Contact Representations and Duality.
DOI: 10.1007/9783642184697_24 Conference: Graph Drawing  18th International Symposium, GD 2010, Konstanz, Germany, September 2124, 2010. Revised Selected Papers
Source: DBLP

Conference Paper: Proportional Contact Representations of Planar Graphs.
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ABSTRACT: We study contact representations for planar graphs, with vertices represented by simple polygons and adjacencies represented by pointcontacts or sidecontacts between the corresponding polygons. Specifically, we consider proportional contact representations, where prespecified vertex weights must be represented by the areas of the corresponding polygons. Several natural optimization goals for such representations include minimizing the complexity of the polygons, the cartographic error, and the unused area. We describe constructive algorithms for proportional contact representations with optimal complexity for general planar graphs and planar 2segment graphs, which include maximal outerplanar graphs and partial 2trees.Graph Drawing  19th International Symposium, GD 2011, Eindhoven, The Netherlands, September 2123, 2011, Revised Selected Papers; 01/2011  [Show abstract] [Hide abstract]
ABSTRACT: In this paper, we consider the problem of representing graphs by polygons whose sides touch. We show that at least six sides per polygon are necessary by constructing a class of planar graphs that cannot be represented by pentagons. We also show that the lower bound of six sides is matched by an upper bound of six sides with a lineartime algorithm for representing any planar graph by touching hexagons. Moreover, our algorithm produces convex polygons with edges having at most three slopes and with all vertices lying on an O(n)xO(n) grid.Algorithmica 04/2011; 63(3). · 0.49 Impact Factor 
Article: Equilateral LContact Graphs
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ABSTRACT: We consider {\em Lgraphs}, that is contact graphs of axisaligned Lshapes in the plane, all with the same rotation. We provide several characterizations of Lgraphs, drawing connections to Schnyder realizers and canonical orders of maximally planar graphs. We show that every contact system of L's can always be converted to an equivalent one with equilateral L's. This can be used to show a stronger version of a result of Thomassen, namely, that every planar graph can be represented as a contact system of squarebased cuboids. We also study a slightly more restricted version of equilateral Lcontact systems and show that these are equivalent to homothetic triangle contact representations of maximally planar graphs. We believe that this new interpretation of the problem might allow for efficient algorithms to find homothetic triangle contact representations, that do not use Schramm's monster packing theorem.03/2013;
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