I have used figures like the one above (left) in my work at various times. It present various distributions in the form of a boxplot, and uses differing labels (in this case, the lowercase letters) to denote significant differences; i.e. levels sharing a label are not significantly different. This type of presentation is common when showing changes in organism condition indices over time (e.g. Figs 3 & 4, Bullseye puffer fish in Mexico).
In the example above, a Kruskal-Wallis rank sum test is used to test differences across all levels, followed by pairwise Mann-Whitney rank tests. The result is a matrix of p-values showing significant differences in distribution. So far so good, but it's not always clear how the grouping relationships should be labelled. In this relatively simple example, the tricky part is that level 1 should be grouped with 3 and 5, but 3 and 5 should not be grouped; Therefore, two labeling codes should be designated, with level 1 sharing both. I have wondered, for some time, if there might be some way to do this in an automated fashion using an algorithm. After many attempts on my own, I finally decided to post a question to SO.
So, my first question "Algorithm for automating pairwise significance grouping labels in R" led me to the concept of the "clique cover problem", and "graph theory" in general, via SO user "David Eisenstat". While I didn't completely understand his recommendation at first, it got me pointed in the right direction - I ultimately found the R package igraph for analyzing and plotting these types of problems.
The next questions were a bit more technical. I figured out that I could return the "cliques" of my grouping relationships network using the cliques function of the igraph package, but my original attempt was giving me a list all relationships in my matrix. It was obvious to me that I would need to identify groupings where all levels were fully connected (i.e. each node in the clique connects to all others). So, my next question "How to identify fully connected node clusters with igraph [in R]" got me a tip from SO user "majom", who showed me that these fully connected cliques could be identified by first reordering the starting nodes in my list of connections (before use in the graph.data.frame function), and then subjecting the resulting igraph object to the function maximal.cliques. So, the first suggestions from David were right on, even though they didn't include code. The result shows nicely all those groupings in the above example (right plot) with fully connected cliques [i.e. (1, 3), (1, 5), (2), (4, 6), (7)].
The final piece of the puzzle was more cosmetic - "How to order a list of vectors based on the order of values contained within [in R]". A bit vague, I know, but what I was trying to do was to label groups in a progressive way so that earlier levels received their labels first. I think this leads to more legible labeling, especially when levels represent some process of progression. At the time of this posting, I have received a single negative (-1) vote on this question... This may have to do with the clarity of the question - I seem to have confused some of the respondents based on follow up comments for clarification - or, maybe someone thought I hadn't shown enough effort on my own. There's no way to know without an accompanying comment. In any case, I got a robust approach from SO user "MrFlick", and I can safely say that I would never have come up with such an eloquent solution on my own.
In all, this solution seems to work great. I have tried it out on larger problems involving more levels and it appears to give correct results. Here is an example with 20 levels (a problem that would have been an amazing headache to do manually):
Code to reproduce the example: