Breadth-First Search in Succinct Planar Graphs

We present a succinct encoding of planar graphs that supports executing a breadth-first search directly on the encoding. The succinct encoding can be constructed in expected $O(n)$ time using $O(n)$ bits during construction; a compact variant can be constructed in deterministic $O(n)$ time using $O(n)$ bits. Once the encoding is constructed, a BFS from any start vertex can be computed in $O(n)$ time using $o(n)$ additional bits, including the space needed to represent the BFS tree. The resulting BFS tree $T$ remains available for standard tree operations, such as traversal, parent and child queries, layer queries, and lowest common ancestor queries, in constant time per query or output element. The encoding also supports standard graph queries. For plane graphs $G=(V, E)$, we provide traversal of the interdigitating tree $\hat T$, i.e., the spanning tree of the dual graph whose edges correspond to $E \setminus E(T)$.
As our main application, we implement the well-known planar separator theorem in a space-efficient way. For biconnected plane graphs, our encoding allows us to compute a balanced separator of size $O(\sqrt n)$ in $O(n)$ time using $o(n)$ additional bits. Along the way, we show that biconnected plane graphs encoded by our representation can be triangulated in expected $O(n)$ time and $o(n)$ bits in the succinct variant, or in deterministic $O(n)$ time using $O(n)$ bits in the compact variant. Further applications include computation of a tree decomposition of width $O(d)$ where $d$ is the diameter of the plane graph at hand and testing for bipartiteness. Finally, all results that do not rely on a plane embedding generalize to separable graph classes.