Given a cell complex K whose geometric realization |K| is embedded in R ³ and a continuous function h: |K|→R (called the height function), we construct a graph G _h (K) which is an extension of the Reeb graph R _h (|K|). More concretely, the graph G _h (K) without loops is a subdivision of R _h (|K|). The most important difference between the graphs G _h (K) and R _h (|K|) 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 R _h (|K|). Moreover, we construct a map ψ: 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 ψ(e) is a cycle in K such that all the points in |ψ(e)| belong to the same level set in |K|.