Homothetic Triangle Contact Representations of Planar Graphs

In this paper we study the problem of computing homothetic triangle contact representations of planar graphs. Since not all planar graphs admit such a representation, we concentrate on meaningful subfamilies of planar graphs and prove that: (i) every two-terminal seriesparallel digraph has a homothetic triangle contact representation, which can be computed in linear time; (ii) every partial planar 3-tree admits a homothetic triangle contact representation.

[1]  J. Pach,et al.  Representation of planar graphs by segments , 1994 .

[2]  Patrice Ossona de Mendez,et al.  On Triangle Contact Graphs , 1994, Combinatorics, Probability and Computing.

[3]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[4]  David G. Kirkpatrick,et al.  Unit disk graph recognition is NP-hard , 1998, Comput. Geom..

[5]  Jeremy P. Spinrad,et al.  Efficient graph representations , 2003, Fields Institute monographs.

[6]  Eugene L. Lawler,et al.  The recognition of Series Parallel digraphs , 1979, SIAM J. Comput..

[7]  Michael Kaufmann,et al.  Max-tolerance graphs as intersection graphs: cliques, cycles, and recognition , 2006, SODA '06.

[8]  Jorge Urrutia,et al.  Discrete Realizations of Contact and Intersection Graphs , 1997, Graph Drawing.

[9]  Petr Hlinený.  Classes and Recognition of Curve Contact Graphs, , 1998, J. Comb. Theory, Ser. B.

[10]  Peter Eades,et al.  Nearest Neighbour Graph Realizability is NP-hard , 1995, LATIN.

[11]  Michael Kaufmann,et al.  On the maximal cliques in c-max-tolerance graphs and their application in clustering molecular sequences , 2006, Algorithms for Molecular Biology.