Shorter Implicit Representation for Planar Graphs and Bounded Treewidth Graphs

Abstract

Implicit representation of graphs is a coding of the structure of graphs using distinct labels so that adjacency between any two vertices can be decided by inspecting their labels alone. All previous implicit representations of planar graphs were based on the classical three forests decomposition technique (a.k.a. Schnyder's trees), yielding asymptotically to a 3 log n-bit label representation where n is the number of vertices of the graph. We propose a new implicit representation of planar graphs using asymptotically 2 log n-bit labels. As a byproduct we have an explicit construction of a graph with n2+o(1) vertices containing all n-vertex planar graphs as induced subgraph, the best previous size of such induced-universal graph was O(n3). More generally, for graphs excluding a fixed minor, we construct a 2 log n + O(log log n) implicit representation. For treewidth-k graphs we give a log n + O(k log log(n/k)) implicit representation, improving the O(k log n) representation of Kannan, Naor and Rudich [18] (STOC '88). Our representations for planar and treewidth-k graphs are easy to implement, all the labels can be constructed in O(n log n) time, and support constant time adjacency testing.