An Information-Theoretic Upper Bound on Planar Graphs Using Well-Orderly Maps
نویسندگان
چکیده
This chapter deals with compressed coding of graphs. We focus on planar graphs, a widely studied class of graphs. A planar graph is a graph that admits an embedding in the plane without edge crossings. Planar maps (class of embeddings of a planar graph) are easier to study than planar graphs, but as a planar graph may admit an exponential number of maps, they give little information on graphs. In order to give an information-theoretic upper bound on planar graphs, we introduce a definition of a quasi-canonical embedding for planar graphs: well-orderly maps. This appears to be an useful tool to study and encode planar graphs. We present upper bounds on the number of unlabeled1 planar graphs and on the number of edges in a random planar graph. We also present an algorithm to compute wellorderly maps and implying an efficient coding of planar graphs.
منابع مشابه
Planar Graphs, via Well-Orderly Maps and Trees
The family of well-orderly maps is a family of planar maps with the property that every connected planar graph has at least one plane embedding which is a well-orderly map.We show that the number of well-orderly maps with n nodes is at most 2αn+O(log , where α ≈ 4.91. A direct consequence of this is a new upper bound on the numberp(n) of unlabeled planar graphs with n nodes, log2 p(n) 4.91n. Th...
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