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Pentagon Contact Representations
Author(s) -
Stefan Felsner,
Hendrik Schrezenmaier,
Raphael Steiner
Publication year - 2018
Publication title -
the electronic journal of combinatorics/the journal of combinatorics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.703
H-Index - 52
eISSN - 1097-1440
pISSN - 1077-8926
DOI - 10.37236/7216
Subject(s) - combinatorics , disjoint sets , mathematics , bijection , homothetic transformation , pentagon , triangulation , representation (politics) , lattice (music) , planar graph , graph , discrete mathematics , geometry , physics , politics , political science , acoustics , law
Representations of planar triangulations as contact graphs of a set of internally disjoint homothetic triangles or of a set of internally disjoint homothetic squares have received quite some attention in recent years. In this paper we investigate representations of planar triangulations as contact graphs of a set of internally disjoint homothetic pentagons. Surprisingly such a representation exists for every triangulation whose outer face is a $5$-gon. We relate these representations to five color forests. These combinatorial structures resemble Schnyder woods and transversal structures, respectively. In particular there is a bijection to certain $\alpha$-orientations and consequently a lattice structure on the set of five color forests of a given graph. This lattice structure plays a role in an algorithm that is supposed to compute a contact representation with pentagons for a given graph. Based on a five color forest the algorithm builds a system of linear equations and solves it, if the solution is non-negative, it encodes distances between corners of a pentagon representation. In this case the representation is constructed and the algorithm terminates. Otherwise negative variables guide a change of the five color forest and the procedure is restarted with the new five color forest. Similar algorithms have been proposed for contact representations with homothetic triangles and with squares.

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