Space-Efficient Graph Coarsening with Applications to Succinct Planar Encodings

We present a novel space-efficient graph coarsening technique for n-vertex separable graphs G, in particular for planar graphs, called 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 P we construct a so-called 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, P) is referred to as a cloud decomposition. We call a graph G=(V, E) separable if it admits to an O(n^c)-separator theorem for some constant c < 1 meaning there exists a separator S subset V that partitions V into {A, S, B} such that no vertices of A and B are adjacent in G and neither A nor B contain more than c'n vertices for a fixed constant c' < 1. Due to the last property such separators are called balanced. This famously includes planar graphs, which admit an O(sqrt(n) n)-separator theorem. 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, 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 input graph. This allows us to construct the succinct encoding scheme for planar 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 a tree decomposition for planar graphs 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. Finally, we generalize our cloud decomposition from planar graphs to arbitrary separable graphs.