Succinct Representations of Separable Graphs
We consider the problem of highly space-efficient representation of separable graphs while supporting queries in constant time in the RAM with logarithmic word size. In particular, we show constant-time support for adjacency, degree and neighborhood queries. For any monotone class of separable graphs, the storage requirement of the representation is optimal to within lower order terms.
Separable graphs are those that admit a O(nc)-separator theorem where c < 1. Many graphs that arise in practice are indeed separable. For instance, graphs with a bounded genus are separable. In particular, planar graphs (genus 0) are separable and our scheme gives the first succinct representation of planar graphs with a storage requirement that matches the information-theory minimum to within lower order terms with constant time support for the queries.
We, furthers, show that we can also modify the scheme to succinctly represent the combinatorial planar embedding of planar graphs (and hence encode planar maps).
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