  An isogeny graph is a graph whose vertices are principally polarized abelian
varieties and whose edges are isogenies between these varieties. In his thesis,
Kohel described the structure of isogeny graphs for elliptic curves and showed
that one may compute the endomorphism ring of an elliptic curve defined over a
finite field by using a depth first search algorithm in the graph. In dimension
2, the structure of isogeny graphs is less understood and existing algorithms
for computing endomorphism rings are very expensive. Our setting considers
genus 2 jacobians with complex multiplication, with the assumptions that the
real multiplication subring is maximal and has class number one. We fully
describe the isogeny graphs in that case. Over finite fields, we derive a depth
first search algorithm for computing endomorphism rings locally at prime
numbers, if the real multiplication is maximal. To the best of our knowledge,
this is the first DFS-based algorithm in genus 2.
