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.
Highlights
Many important families of graphs can be defined through a forbidden graph characterization
We try to showcase in this paper, that the split-decomposition is a very convenient tool by which to find induced subpatterns: various connected portions of the graphs may be broken down into far apart blocks in the split-decomposition tree, the property that there is an alternated path between any two vertices that are connected in the original graph is very powerful, and as we show in Section 2 of this paper, allows to deduce constraints following the appearance of an induced pattern or subgraph
We follow the ideas of Gioan and Paul [19] and Chauve et al [8], and provide full analyses of several important subclasses of distance-hereditary graph. Some of these analyses have lead us to uncover previously unknown enumerations, while for other classes for which enumerations were already known, we have provided symbolic grammars which are a more powerful starting point for future work: such as parameter analyses, exhaustive and random generation and the empirical analyses that the latter enables
Summary
Many important families of graphs can be defined (sometimes exclusively) through a forbidden graph characterization. 3. Forbidden induced subgraphs, in which we try to avoid certain induced subgraphs from appearing (that is we pick a subset of vertices, and use all edges with both endpoints in that subset). We try to showcase in this paper, that the split-decomposition is a very convenient tool by which to find induced subpatterns: various connected portions of the graphs may be broken down into far apart blocks in the split-decomposition tree, the property that there is an alternated path between any two vertices that are connected in the original graph is very powerful, and as we show in Section 2 of this paper, allows to deduce constraints following the appearance of an induced pattern or subgraph.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
