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 graphinternal vertexgraph classcontact representationbottom facethese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
|Title of host publication||Graph-Theoretic Concepts in Computer Science. WG 2013|
|Editors||A. Brandstädt, A. Jansen, R. Reischuk|
|Publication status||Published - 2013|
|Series||Lecture Notes in Computer Science|