A fast general methodology for information — theoretically optimal encodings of graphs

We propose a fast methodology for encoding graphs with information-theoretically minimum numbers of bits. The methodology is applicable to general classes of graphs; this paper focuses on simple planar graphs. Specifically, a graph with property π is called a π-graph.If π satisfies certain properties, then an n-node π-graph G can be encoded by a binary string X such that (1) G and X can be obtained from each other in O(n log n) time, and (2) X has at most β(n)+o(β(n)) bits for any function β(n) = Ω{n) so that there are at most 2^{β(n)+o(β (n))} distinct n-node n-graphs. Examples of such n include all conjunctions of the following sets of properties: (1) G is a planar graph or a plane graph; (2) G is directed or undirected; (3) G is triangulated, triconnected, biconnected, merely connected, or not required to be connected; and (4) G has at most ℓ₁ (respectively, ℓ₂) distinct node (respectively, edge) labels. These examples are novel applications of small cycle separators of planar graphs and settle several problems that have been open since Tutte's census series were published in 1960's.