Enumerations, Forbidden Subgraph Characterizations, and the Split-Decomposition

Bahrani, Maryam; Lumbroso, Jérémie

doi:10.48550/arxiv.1608.01465

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“…In this section, we introduce standard definitions from graph theory (1.1), followed by a formal introduction to the splitdecomposition expressed in terms of graph-labeled trees (1.2 and 1.3), which are the tools also used by Chauve et al [11] and Bahrani and Lumbroso [2].…”

Section: Definitions and Preliminariesmentioning

confidence: 99%

“…These graph-labeled trees are a powerful tool for studying the structure of the original graph they describe. Some elements of the terminology have been summarized in Figure 4 (reproduced from Bahrani and Lumbroso [2]), as they are frequently referenced in the proofs of Section 2. DEFINITION 2.…”

Section: Definitions and Preliminariesmentioning

confidence: 99%

“…The paper by Chauve et al [11] was first in a line of work on the application of the split decomposition to graph enumeration. Bahrani and Lumbroso [2] used the same framework to give grammars for subsets of distancehereditary graphs that are specified in terms of forbidden induced subgraphs. They showed that constraining a split decomposition tree to avoid certain patterns can help avoid corresponding induced subgraphs in the original graph.…”

Section: Introductionmentioning

confidence: 99%

“…distance-hereditary graphs and their subfamilies. As stated by Bahrani and Lumbroso [2], the natural next step to develop its reach is to study split decomposition trees that are not fully decomposable and contain prime nodes. This is part of the motivation behind studying cactus graphs, which result in split decomposition trees with prime polygonal nodes.…”

Section: Introductionmentioning

confidence: 99%

“…Since cactus graphs are very similar to trees, the same grammars can be developed more directly by exploiting this apparent recursivity. However, while the entire machinery introduced in this paper is not necessary to obtain grammars for cactus graphs, these tools belong to a more general and powerful framework developed by Chauve et al [11] and extended by Bahrani and Lumbroso [2]. This framework is highly flexible and can be generalized to obtain grammars for many other classes of cactus graphs.…”

Section: Introductionmentioning

confidence: 99%

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

Bahrani¹,

Lumbroso²

2018

2018 Proceedings of the Fifteenth Workshop on Analytic Algorithmics and Combinatorics (ANALCO)

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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 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.

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Section: Definitions and Preliminariesmentioning

confidence: 99%

Section: Definitions and Preliminariesmentioning

confidence: 99%