# Normalising graphs of groups

### (16 pages)

**Abstract.**
We discuss a partial
normalisation of a finite graph of finite groups (*\Gamma*(-), *X*)
which leaves invariant the fundamental group. In conjunction with an
easy graph-theoretic result, this provides a flexible and rather
useful tool in the study of finitely generated virtually free
groups. Applications discussed here include (i) an important
inequality for the number of edges in a Stallings decomposition
*\Gamma* ~ *\pi*_{1}(*\Gamma*(-), *X*) of a finitely generated virtually
free group, (ii) the proof of equivalence of a number of conditions
for such a group to be `large', as well as (iii) the classification up
to isomorphism of virtually free groups of (free) rank 2. We also
discuss some number-theoretic consequences of the last result.

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