Split-Decomposition Trees with Prime Nodes: Enumeration and Random Generation of Cactus Graphs

TL;DR

This paper characterizes the split-decomposition trees of cactus graphs, develops symbolic grammars for their enumeration, and implements a method for their random generation, extending combinatorial graph analysis techniques.

Contribution

It provides a new characterization and enumeration framework for cactus graphs using split-decomposition trees with prime nodes, and introduces a random generation method.

Findings

Derived a characterization of split-decomposition trees of cactus graphs.

Developed symbolic grammars for enumerating cactus graphs.

Implemented a random generation algorithm for cactus graphs.

Abstract

In this paper, we build on recent results by Chauve et al. (2014) and Bahrani and Lumbroso (2017), which combined the split-decomposition, as exposed by Gioan and Paul, with analytic combinatorics, to produce new enumerative results on graphs---in particular the enumeration of several subclasses of perfect graphs (distance-hereditary, 3-leaf power, ptolemaic). Our goal was to study a simple family of graphs, of which the split-decomposition trees have prime nodes drawn from an enumerable (and manageable!) set of graphs. Cactus graphs, which we describe in more detail further down in this paper, can be thought of as trees with their edges replaced by cycles (of arbitrary lengths). Their split-decomposition trees contain prime nodes that are cycles, making them ideal to study. We derive a characterization for the split-decomposition trees of cactus graphs, produce a general template of symbolic grammars for cactus graphs, and implement random generation for these graphs, building on work by Iriza (2015).