Canonical Decomposition of Outerplanar Maps and Application to Enumeration, Coding and Generation

Abstract

In this article we define a canonical decomposition of rooted outerplanar maps into a spanning tree and a list of edges. This decomposition, constructible in linear time in the Word-RAM model, implies the existence of bijection between rooted outerplanar maps with n nodes and bicolored rooted ordered trees with n nodes where all the nodes of the last branch are colored white. As a consequence, for rooted outerplanar maps of n nodes, we derive:

• an enumeration formula, and an asymptotic of 2^{3n −Θ(logn)};
• an optimal data structure of asymptotically 3n bits, built in O(n) time, supporting adjacency and degree queries in worst-case constant time and neighbors query of a degree-d node in worst-case O(d) time.
• an O(n) expected time uniform random generating algorithm.