A Graph-with-Loop Structure for a Topological Representation of 3D Objects ⋆ Rocio Gonzalez-Diaz, Mar´ ıa Jos´ e Jim´ enez, Belen Medrano ⋆⋆ , and Pedro Real Applied Math Department, University of Seville, Spain {rogodi,majiro,belenmg,real}@us.es http://alojamientos.us.es/gtocoma Abstract. 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|. 1 Reeb Graphs and Tunnels We are interested in analyzing and visualizing intrinsic properties of geometric models and scientific data. Specifically, Reeb graphs [13], which express the con- nectivity of level sets, have been used in the past to construct data structures and user-interfaces for modeling and visualization applications [5]. Let X be a topological space and h : X → R a continuos map. A level set is the primage of a constant value, h -1 (t). Call a connected component of a level set a contour. Two points x, y ∈ X are equivalent, x ∼ y, if they belong to the same contour, that is, if h(x)= h(y) and x and y are connected by a path on X . The Reeb graph of h, R h (X ), is the quotient space defined by this equivalence relation. Observe that, by construction, the Reeb graph has a point for each contour and the connection is provided by ψ : X → R h (X ) that maps each point x to its equivalence class. Even though the Reeb graph loses a lot of the original topological structure, some things can be said: a tunnel in X that maps (by ψ) to a tunnel in R h (X ) cannot be continuously deformed to a single point, and two tunnels in X that map to different tunnels in R h (X ). The number of connected components of X , β 0 (X ), is preserved and the number of tunnels of X , β 1 (X ), cannot increase, i.e. β 0 (R h (X )) = β 0 (X ) and β 1 (R h (X )) ≤ β 1 (X ). ⋆ Partially supported by Junta de Andaluc´ ıa (FQM-296 and TIC-02268) and Spanish Ministry for Science and Education (MTM-2006-03722). ⋆⋆ Fellow associated to University of Seville under a Junta de Andalucia research grant. W.G. Kropatsch, M. Kampel, and A. Hanbury (Eds.): CAIP 2007, LNCS 4673, pp. 506–513, 2007. c  Springer-Verlag Berlin Heidelberg 2007