Enumerations, Forbidden Subgraph Characterizations, and the Split-Decomposition

Abstract

Forbidden characterizations may sometimes be the most natural way to describe families of graphs, and yet these characterizations are usually very hard to exploit for enumerative purposes.By building on the work of Gioan and Paul (2012) and Chauve et al. (2014), we show a methodology by which we constrain a split-decomposition tree to avoid certain patterns, thereby avoiding the corresponding induced subgraphs in the original graph.We thus provide the grammars and full enumeration for a wide set of graph classes: ptolemaic, block, and variants of cactus graphs (2,3-cacti, 3-cacti and 4-cacti). In certain cases, no enumeration was known (ptolemaic, 4-cacti); in other cases, although the enumerations were known, an abundant potential is unlocked by the grammars we provide (in terms of asymptotic analysis, random generation, and parameter analyses, etc.).We believe this methodology here shows its potential; the natural next step to develop its reach would be to study split-decomposition trees which contain certain prime nodes. This will be the object of future work.

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

“…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.…”

“…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].…”

“…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 Fig [2]) illustrate the concept of an alternated path: it is, more informally, a path that only ever uses at most one interior edge of any graph label.…”

“…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.…”

“…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 (a) A random mixed cactus graph with 309 vertices and 80 cycles.…”