Enumerating Rooted Graphs with Reflectional Block Structures

Abstract

In this paper, we consider an arbitrary class ${\cal H}$ of rooted graphs such that each biconnected component is given by a representation with reflectional symmetry, which allows a rooted graph to have several different representations, called embeddings. We give a general framework to design algorithms for enumerating embeddings of all graphs in ${\cal H}$ without repetition. The framework delivers an efficient enumeration algorithm for a class ${\cal H}$ if the class ${\cal B}$ of biconnected graphs used in the graphs in ${\cal H}$ admits an efficient enumeration algorithm. For example, for the class ${\cal B}$ of rooted cycles, we can easily design an algorithm of enumerating rooted cycles so that delivers the difference between two consecutive cycles in constant time in a series of all outputs. Hence our framework implies that, for the class ${\cal H}$ of all rooted cacti, there is an algorithm that enumerates each cactus in constant time.