Word game for those sleepless nights

I'm an enthusiastic solver of cryptic crossword puzzles and also have a keen interest in words (their spelling, meaning and origin).  I don't know which interest came first but they seem to reinforce one another.  In particular I love anagrams. I was tickled pink recently listening to an episode of the BBC's News Quiz which touched on the story last year of the Duchess of Cambridge being photographed topless: I learnt that 'Kate Middleton' is an anagram of 'Naked Tit Model'.

Anyway that is a bit off-track for this post, which is about a solitaire word game based on anagrams.  You take any word you like and remove its letters one by one (in any order you like) and, at each stage, you have to make an anagram of the current set of letters.  For example, if you start with 'wonderer' you could proceed as drowner, wonder, drown, down, own, no, o.  Such a word is deemed to be a success, otherwise it is a fail.

With practice the game can be played in your head and the more experienced you get the longer the word you can start with.  I find words of 9 or more letters a bit of a problem but I've discovered many 8 letter successes.

Here are some more examples of successful words to give you the idea:
many, man, an, a
words, word, row, or, o
demand, named, name, man, an, a
diligent, tingled, tingle, glint, lint, tin, in, i
efforts, forest, store, rote, toe, to, o

Obviously, to be a success the word has to contain at least one of the letters a, i, o (the only one-letter words).  A stronger necessary condition for success (but not so easily checked) is that the set of letters you start with must have letter subsets of all sizes which have anagrams.  For example 'ice' is a failure because it has no two-letter anagrams.  On the other hand

It is astonishing how many successful words there are (especially if you restrict yourself to words satisfying these necessary conditions).  When trying to "solve a word" I find it easiest to begin with the word itself and reduce it to one letter as in the examples above.  This would be 'top-down' approach.  But I suspect that a bottom-up approach - building up longer and longer words in several ways - would be a more efficient approach for a computer program; a natural use for the approach called 'dynamic programming'.

There are several ways in which one can 'mathematize' the basic idea.  For example one could define a graph whose vertices are English words and where there was a directed edge from one word to another if the second word was an anagram of the first word without one of its letters.  Then one might ask about the connected components of this graph.

But you don't have to be a mathematician to enjoy the game.  Have a go yourself (ourself, flours, flour, four, for, of, o).