  A Coxeter system is an ordered pair (W,S) where S is the generating set in a
particular type of presentation for the Coxeter group W. A subgroup of W is
called special if it is generated by a subset of S. Amalgamated product
decompositions of a Coxeter group having special factors and special
amalgamated subgroup are easily recognized from the presentation of the Coxeter
group. If a Coxeter group is a subgroup of the fundamental group of a given
graph of groups, then the Coxeter group is also the fundamental group of a
graph of special subgroups, where each vertex and edge group is a subgroup of a
conjugate of a vertex or edge group of the given graph of groups. A vertex
group of an arbitrary graph of groups decomposition of a Coxeter group is shown
to split into parts conjugate to special groups and parts that are subgroups of
edge groups of the given decomposition. Several applications of the main
theorem are produced, including the classification of maximal FA-subgroups of a
finitely generated Coxeter group as all conjugates of certain special
subgroups.
