Just to clearify notions: You speak about the Tits Buildings and say their theory (which part?) is only that much help, if it's "thick", i.e. each residue has at least three chambers? (In contrast a Coxeter building, which is what the "appartments" are, has only exactly two)

I'll have a look at the paper you suggested, thanx! Well, the term already appears in appendix F of Aschbachers highly influencial "Classification of Quasi-Thin Groups" (As I understand that's the topic on which he closed the classification theorem?) But I have NO-CLUE what he's talking about and I can't see the connection e.g. to the introduction "Buildings" by Brian Lehmann. (HELP ;-) ) Also, it's certainly no accidient, the BN-pair of the Monster is named that way and exactly looks like the one of a doubly transitive group? And this seems like THE road to the monster?

No, the dodecahedral graph is just 3-regular (hence 30 edges) and is planar (I didn't compute "my" genus so far). The GIRTH is the SHORTEST cycle (here indeed barely 3, triangle-free would mean >3), while the diameter is the longest shortest ;-) path.

Thanks alot ... I should've though of using a computer tool ;-) If someone reongnizes, I would still be interested, if this is a "known" graph...