Short encodings of planar graphs and maps
Discrete Applied Mathematics
Linear-Time Succinct Encodings of Planar Graphs via Canonical Orderings
SIAM Journal on Discrete Mathematics
Compact Encodings of Planar Graphs via Canonical Orderings and Multiple Parentheses
ICALP '98 Proceedings of the 25th International Colloquium on Automata, Languages and Programming
Succinct representation of balanced parentheses, static trees and planar graphs
FOCS '97 Proceedings of the 38th Annual Symposium on Foundations of Computer Science
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We propose a fast methodology for encoding graphs with informationtheoretically 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 驴 is called a 驴-graph. If 驴 satisfies certain properties, then an n-node 驴-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 脽(n)+o(脽(n)) bits for any function 脽(n) = 驴(n) so that there are at most 2脽(n)+o(脽(n)) distinct n-node 驴-graphs. Examples of such 驴 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.