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

@inproceedings{Bahrani2018SplitDecompositionTW, title={Split-Decomposition Trees with Prime Nodes: Enumeration and Random Generation of Cactus Graphs}, author={Maryam Bahrani and J{\'e}r{\'e}mie O. Lumbroso}, booktitle={ANALCO}, year={2018} }

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…

## Figures and Tables from this paper

## 3 Citations

### Enumerations, Forbidden Subgraph Characterizations, and the Split-Decomposition

- Mathematics, Computer ScienceElectron. J. Comb.
- 2018

A methodology by which a split-decomposition tree is constrain to avoid certain patterns, thereby avoiding the corresponding induced subgraphs in the original graph is shown.

### Exact-size Sampling of Enriched Trees in Linear Time

- Mathematics, Computer Science
- 2021

This work constructs expected linear time samplers for critical Bienaymé–Galton–Watson trees having exactly n nodes with outdegree in some fixed set, enabling uniform generation for many combinatorial classes such as dissections of polygons.

### Random multi-hooking networks

- Computer Science
- 2022

The degree proﬁle in random multi-hooking networks is analyzed by track-ing two kinds of node degrees—the local average degree of a speci⬁c node over time and the global overall average degree in the graph.

## References

SHOWING 1-10 OF 34 REFERENCES

### Enumerations, Forbidden Subgraph Characterizations, and the Split-Decomposition

- Mathematics, Computer ScienceElectron. J. Comb.
- 2018

A methodology by which a split-decomposition tree is constrain to avoid certain patterns, thereby avoiding the corresponding induced subgraphs in the original graph is shown.

### Split decomposition and graph-labelled trees: characterizations and fully-dynamic algorithms for totally decomposable graphs

- Mathematics, Computer ScienceDiscret. Appl. Math.
- 2012

### An Exact Enumeration of Distance-Hereditary Graphs

- MathematicsANALCO
- 2017

The power of revisiting graph decomposition results through the framework of analytic combinatorics is illustrated, which allows to show that the number of distance-hereditary graphs on $n$ vertices is tightly bounded by ${(7.24975\ldots)^n}$---opening the perspective such graphs could be encoded on $3n$ bits.

### Enumeration and Random Generation of Unlabeled Classes of Graphs: A Practical Study of Cycle Pointing and the Dissymmetry Theorem

- MathematicsArXiv
- 2015

The power of the dissymmetry theorem is extended by showing that it in fact provides a Boltzmann sampler for the class in question, and an exposition of the cycle pointing technique is presented, with a focus on the enumeration and random generation of the underlying unpointed class.

### A Calculus for the Random Generation of Labelled Combinatorial Structures

- Computer Science, MathematicsTheor. Comput. Sci.
- 1994

### Combinatorial species and tree-like structures

- Computer ScienceEncyclopedia of mathematics and its applications
- 1997

The combinatorial theory of species, introduced by Joyal in 1980, provides a unified understanding of the use of generating functions for both labelled and unlabelled structures and as a tool for the…

### Enumeration of m-Ary Cacti

- MathematicsAdv. Appl. Math.
- 2000

The purpose of this paper is to enumerate various classes of cyclically colored m-gonal plane cacti, called m-ary cactu, according to the topological classification of complex polynomials having at most m critical values, studied by Zvonkin and others.

### Parameterized complexity of finding a spanning tree with minimum reload cost diameter

- MathematicsIPEC
- 2017

It is proved that DIAMETER-TREE is para-NP-hard for any combination of two of these three parameters, and that it is FPT parameterized by the three of them.

### Decomposition of Directed Graphs

- Mathematics
- 1982

A composition for directed graphs which generalizes the substitution (or X-join) composition of graphs and digraphs, as well as the graph version of set-family composition, is described. It is proved…