Navigating planar topologies in near-optimal space and time

We show that any embedding of a planar graph can be encoded succinctly while efficiently answering a number of topological queries near-optimally. More precisely, we build on a succinct representation that encodes an embedding of m edges within 4m bits, which is close to the information-theoretic lower bound of about 3.58m. With 4m+o(m) bits of space, we show how to answer a number of topological queries relating nodes, edges, and faces, most of them in any time in ω(1). Indeed, 3.58m+o(m) bits suffice if the graph has no self-loops and no nodes of degree one. Further, we show that with O(m) bits of space we can solve all those operations in O(1) time.