Why modular decomposition is not very useful for chemical graphs

It is difficult to publish negative results in a journal, but a blog post seems like a good place to record the experience. Especially situations like this, where it probably should have been obvious not to try in the first place...

So; what is modular decomposition? Briefly, a module is a little like a connected component in a graph - indeed, a connected component is made up from one or more modules, but modules can overlap. Decomposition of a graph into its modules is, therefore, like finding the connected components of the graph. An example is shown here:

Two modules in the graph are circled, there may be others. The definition is a set of vertices that have the same neighbours outside the set. So, there was no need for me to make them complete graphs, but it looked nicer. Anyway, already looking at this example it is clear that these are not very 'chemical' graphs. They look more like networks (for example, see : [1]).

Indeed, I tried out some code made by the authors of a paper on a linear-time algorithm for modular decomposition [2] that was in java (makes a change from c). It was also nicely commented, unlike the mathematician - whose name I won't mention - that said he doesn't comment or document his code because "no program has ever improved through comments" which is just a lazy excuse, frankly.

The results for the molecules in the CDK MoleculeFactory were that almost all of them are prime modules; which means that they are elementary, or unbreakdownable. Notable exceptions are cyclobutane and a propellane-like graph (see image, modules are circled).

In fact, I suspect that chemical graphs with non-prime modular decomposition trees are rare. Partly because most graphs are irregular, but mainly due to the low degree (valence) of atomic vertices. Anyway, modules are not a solution to structure diagram layout [3].

[1] : J. Gagneur, R. Krause, T. Bouwmeester, and G. Casari. Modular decomposition of protein-protein interaction networks. Genome Biology 5:R57 (2004). doi:10.1186/gb-2004-5-8-r57

[2] : Marc Tedder, Derek Corneil, Michel Habib, Christophe Paul Simple. Simpler linear-time modular decomposition via recursive factorizing permutations. DOI:10.1007/978-3-540-70575-8_52

[3] Drawing Graphs Using Modular Decomposition DOI: 10.1007/11618058_31

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