A succinct encoding of planar graphs supports BFS in O(n) time with o(n) extra bits, keeps the BFS tree queryable, and yields sublinear-space planar separator and bipartiteness algorithms.
Sublinear-Space Lexicographic Depth-First Search for Bounded Treewidth Graphs and Planar Graphs
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Breadth-First Search in Succinct Planar Graphs
A succinct encoding of planar graphs supports BFS in O(n) time with o(n) extra bits, keeps the BFS tree queryable, and yields sublinear-space planar separator and bipartiteness algorithms.