  Reeb graphs are structural descriptors that capture shape properties of a
topological space from the perspective of a chosen function. In this work we
define a combinatorial metric for Reeb graphs of orientable surfaces in terms
of the cost necessary to transform one graph into another by edit operations.
The main contributions of this paper are the stability property and the
optimality of this edit distance. More precisely, the stability result states
that changes in the functions, measured by the maximum norm, imply not greater
changes in the corresponding Reeb graphs, measured by the edit distance. The
optimality result states that our edit distance discriminates Reeb graphs
better than any other metric for Reeb graphs of surfaces satisfying the
stability property.
