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 $\omega (1)$. Further, we show that with $O(m)$ bits of space we can solve all those operations in $O(1)$ time.