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

Maryam Bahrani, Jérémie Lumbroso
2018 2018 Proceedings of the Fifteenth Workshop on Analytic Algorithmics and Combinatorics (ANALCO)  
In this paper, we build on recent results by Chauve et al. and Bahrani and Lumbroso, which combined the splitdecomposition, 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
more » ... 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 (a) A random mixed cactus graph with 309 vertices and 80 cycles. (b) A random mixed cactus graph with 933 vertices and 239 cycles. Figure 1. Our symbolic grammars can be used to create Boltzmann samplers for the uniform random generation of large cactus graphs. Here we have included two cactus graphs, each of which has been drawn uniformly at random from the family of unlabeled, plane, rooted cacti where each cycle has at least 4 nodes.
doi:10.1137/1.9781611975062.13 dblp:conf/analco/BahraniL18 fatcat:uupao4f4yfhvfkif7fpkarb7yq