### Abstract

We propose a fast methodology for encoding graphs with information-theoretically minimum numbers of bits. Specifically, a graph with property π is called a π-graph. If π satisfies certain properties, then an n-node m-edge 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 continuous superadditive function β(n) so that there are at most 2^{β(n)+o(β(n))} distinct n-node π-graphs. The methodology is applicable to general classes of graphs; this paper focuses on planar graphs. Examples of such π include all conjunctions over the following groups 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; (4) the nodes of G are labeled with labels from {1, . . . , ℓ_{1}} for ℓ_{1} ≤ n; (5) the edges of G are labeled with labels from {1, . . . , ℓ_{2}} for ℓ_{2} ≤ m; and (6) each node (respectively, edge) of G has at most ℓ_{3} = O(1) self-loops (respectively, ℓ_{4} = O(1) multiple edges). Moreover, ℓ_{3} and ℓ_{4} are not required to be O(1) for the cases of π being a plane triangulation. These examples are novel applications of small cycle separators of planar graphs and are the only nontrivial classes of graphs, other than rooted trees, with known polynomial-time information-theoretically optimal coding schemes.

Original language | English (US) |
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Pages (from-to) | 838-846 |

Number of pages | 9 |

Journal | SIAM Journal on Computing |

Volume | 30 |

Issue number | 3 |

DOIs | |

State | Published - Jan 1 2000 |

### Keywords

- Biconnected graphs
- Cycle separators
- Data compression
- Graph encoding
- Planar graphs
- Triangulations
- Triconnected graphs

### ASJC Scopus subject areas

- Computer Science(all)
- Mathematics(all)

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## Cite this

*SIAM Journal on Computing*,

*30*(3), 838-846. https://doi.org/10.1137/S0097539799359117