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

摘要

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, 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: 1. an enumeration formula, and an asymptotic of 2~(3n-Θ(log n)); 2. an optimal data structure of asymptotically 3n bits, built in O(n) time, supporting adjacency and degree queries in worst-case constant time; 3. an O(n) expected time uniform random generating algorithm.