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 $4m+o\left(m\right)$ 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 $\omega \left(1\right)$. Indeed, $3.58m+o\left(m\right)$ bits suffice if the graph has no self-loops and no nodes of degree one. Further, we show that with $O\left(m\right)$ bits of space we can solve all those operations in $O\left(1\right)$ time.