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 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 ? 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, ?) is referred to as a 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, ?) 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 ?(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 + ?}) for any ? > 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 ?n log n) for graphs of treewidth k ? ?n in sublinear space and polynomial time

On the Enumeration of all Minimal Triangulations

We present an algorithm that enumerates all the minimal triangulations of a graph in incremental polynomial time. Consequently, we get an algorithm for enumerating all the proper tree decompositions, in incremental polynomial time, where "proper" means that the tree decomposition cannot be improved by removing or splitting a bag

Compact data structures for triangulations

International audienceThe main problem consists in designing space-efficient data structures allowing to represent the connectivity of triangle meshes while supporting fast navigation and local updates

Distance Oracles for Interval Graphs via Breadth-First Rank/Select in Succinct Trees

We present the first succinct distance oracles for (unweighted) interval graphs and related classes of graphs, using a novel succinct data structure for ordinal trees that supports the mapping between preorder (i.e., depth-first) ranks and level-order (breadth-first) ranks of nodes in constant time. Our distance oracles for interval graphs also support navigation queries – testing adjacency, computing node degrees, neighborhoods, and shortest paths – all in optimal time. Our technique also yields optimal distance oracles for proper interval graphs (unit-interval graphs) and circular-arc graphs. Our tree data structure supports all operations provided by different approaches in previous work, as well as mapping to and from level-order ranks and retrieving the last (first) internal node before (after) a given node in a level-order traversal, all in constant time

Distance Oracles for Interval Graphs via Breadth-First Rank/Select in Succinct Trees

We present the first succinct distance oracles for (unweighted) interval graphs and related classes of graphs, using a novel succinct data structure for ordinal trees that supports the mapping between preorder (i.e., depth-first) ranks and level-order (breadth-first) ranks of nodes in constant time. Our distance oracles for interval graphs also support navigation queries - testing adjacency, computing node degrees, neighborhoods, and shortest paths - all in optimal time. Our technique also yields optimal distance oracles for proper interval graphs (unit-interval graphs) and circular-arc graphs. Our tree data structure supports all operations provided by different approaches in previous work, as well as mapping to and from level-order ranks and retrieving the last (first) internal node before (after) a given node in a level-order traversal, all in constant time