Threshold-Coloring and Unit-Cube Contact Representation of Graphs
We study threshold coloring of graphs where the vertex colors, represented by integers, describe any spanning subgraph of the given graph as follows. Pairs of vertices with near colors imply the edge between them is present and pairs of vertices with far colors imply the edge is absent. Not all planar graphs are threshold-colorable, but several subclasses, such as trees, some planar grids, and planar graphs with no short cycles can always be threshold-colored. Using these results we obtain unit-cube contact representation of several subclasses of planar graphs. We show the NP-completeness for two variants of the threshold coloring problem and describe a polynomial-time algorithm for another.
KeywordsPlanar Graph Internal Vertex Graph Class Contact Representation Bottom Face
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- 1.Alam, M.J., Chaplick, S., Fijavž, G., Kaufmann, M., Kobourov, S.G., Pupyrev, S.: Threshold-coloring and unit-cube contact representation of graphs. ArXiv report (2013), http://arxiv.org/abs/1305.0069
- 2.Brandstädt, A., Le, V.B., Spinrad, J.P.: Graph classes: A survey. Society for Industrial and Applied Mathematics (1999)Google Scholar
- 5.Felsner, S., Francis, M.C.: Contact representations of planar graphs with cubes. In: Symposium on Computational Geometry, pp. 315–320 (2011)Google Scholar
- 10.Mahadev, N.V.R., Peled, U.N.: Threshold Graphs and Related Topics. N. Holland (1995)Google Scholar
- 11.Roberts, F.S.: Indifference graphs. In: Proof Techniques in Graph Theory, pp. 139–146. Academic Press (1969)Google Scholar
- 12.Roberts, F.S.: From garbage to rainbows: Generalizations of graph coloring and their applications. Graph Theory, Combinatorics, and Applications 2, 1031–1052 (1991)Google Scholar