Space-Efficient Graph Coarsening with Applications to Succinct Planar Encodings

We present a novel space-efficient graph coarsening technique for $n$-vertex planar graphs $G$, called \textit{cloud partition}, which partitions the vertices $V(G)$ into disjoint sets $C$ of size $O(\log n)$ such that each $C$ induces a connected subgraph of $G$. Using this partition $\mathcal{P}$ we construct a so-called \textit{structure-maintaining minor} $F$ of $G$ via specific contractions within the disjoint sets such that $F$ has $O(n/\log n)$ vertices. The combination of $(F, \mathcal{P})$ is referred to as a \textit{cloud decomposition}. For planar graphs we show that a cloud decomposition can be constructed in $O(n)$ time and using $O(n)$ bits. Given a cloud decomposition $(F, \mathcal{P})$ constructed for a planar graph $G$ we are able to find a balanced separator of $G$ in $O(n/\log n)$ time. Contrary to related publications, we do not make use of an embedding of the planar input graph. We generalize our cloud decomposition from planar graphs to $H$-minor-free graphs for any fixed graph $H$. This allows us to construct the succinct encoding scheme for $H$-minor-free graphs due to Blelloch and Farzan (CPM 2010) in $O(n)$ time and $O(n)$ bits improving both runtime and space by a factor of $\Theta(\log n)$. As an additional application of our cloud decomposition we show that, for $H$-minor-free graphs, a tree decomposition of width $O(n^{1/2 + \epsilon})$ for any $\epsilon > 0$ can be constructed in $O(n)$ bits and a time linear in the size of the tree decomposition. A similar result by Izumi and Otachi (ICALP 2020) constructs a tree decomposition of width $O(k \sqrt{n} \log n)$ for graphs of treewidth $k \leq \sqrt{n}$ in sublinear space and polynomial time.