A graph-with-loop structure for a topological representation of 3D objects

Given a cell complex K whose geometric realization vertical bar K vertical bar is embedded in R-3 and a continuous function h : vertical bar K vertical bar -> R (called the height function), we construct a graph G(h)(K) which is an extension of the Reeb graph R-h(vertical bar K vertical bar). More concretely, the graph Gh(K) without loops is a subdivision of R-h(vertical bar K vertical bar). The most important difference between the graphs G(h)(K) and R-h(vertical bar K vertical bar) is that G(h)(K) preserves not only the number of connected components but also the number of "tunnels" (the homology generators of dimension 1) of K. The latter is not true in general for Rh(vertical bar K vertical bar). Moreover, we construct a map psi : G(h)(K) K identifying representative cycles of the tunnels in K with the ones in G(h)(K) in the way that if e is a loop in G(h)(K), then psi(e) is a cycle in K such that all the points in vertical bar psi(e)vertical bar belong to the same level set in vertical bar K vertical bar.