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Disc structure of certain chamber graphs

A chamber system over the set I consists of a set C and a system (Pi)i∈I of partitions of C indexed by I. The elements of C are called chambers and, for brevity, the system (C, (Pi)i∈I) will often be referred to as the chamber system C. Two chambers which are both in the same member of Pj for some j ∈ I are said to be adjacent chambers. The chamber graph of the chamber system C is the graph with vertex set C and two (distinct) chambers are adjacent (in the chamber graph) if they are adjacent chambers in C. (Note that a chamber is not adjacent to itself in the chamber graph.) If two chambers c, c are both in the same member of Pi (i ∈ I), then we say they are i-adjacent, and denote this by c ∼i c ′ (or c ∼i c). The rank of the chamber system is the cardinality of I. An automorphism of C is a permutation σ of C which preserves each of the partitions Pi, that is, whenever c ∼i c ′ (c, c ∈ C, i ∈ I) then cσ ∼i c σ. We shall now concentrate on the following situation.