  We develop a general theory of "bisets": sets with two commuting group
actions. They naturally encode topological correspondences. Just as van
Kampen's theorem decomposes into a graph of groups the fundamental group of a
space given with a cover, we prove analogously that the biset of a
correspondence decomposes into a "graph of bisets": a graph with bisets at its
vertices, given with some natural maps. The "fundamental biset" of the graph of
bisets recovers the original biset. We apply these results to decompose the
biset of a Thurston map (a branched self-covering of the sphere whose critical
points have finite orbits) into a graph of bisets. This graph closely parallels
the theory of Hubbard trees. This is the first part of a series of five
articles, whose main goal is to prove algorithmic decidability of combinatorial
equivalence of Thurston maps.
