INTERNATIONAL JOURNAL OF GEOMETRY Vol. 6 (2017), No. 1, 89-94 Note on an Algorithm for Computing the Reeb Graph ADELA LUPESCU Abstract. The Reeb graph of a real function defined on a manifold is a useful mathematical tool with applications in computational topology, com- putational geometry, determination of level sets, computer graphics and so on. The main objective of this paper is to present an algorithm to determine the Reeb graph of a real valued function defined on a compact surface. 1. Introduction Given a topological space X and a real valued function f : X ! R, define on X the following equivalence relation ∼ : we have x ∼ y if and only if x and y belong to the same path connected component of a level set f −1(a), a 2 R. The Reeb graph is the quotient space X=∼ := R(f). Here are few examples. Example 1.1. (1)The Reeb graph of a sphere when f is the height function: |||||||||||||||||{ Keywords and phrases: critical point; Reeb graph; computational topology (2010)Mathematics Subject Classification: 58E05 Received: 5.02.2017 In revised form: 25.03.2017 Accepted: 30.03.2017. 90 Adela Lupescu (2) The Reeb graph of a torus when f is the height function : (3)The Reeb graph of a torus of genus 2 when f is the height function : The Reeb graph, originally defined in the context of Morse theory in the paper [7], can be used to study properties of a space through the lens of a real-valued function by providing a way to track and visualize the connected components of the space at level sets of the function. Recall that, a category C is an algebraic structure consisting of the fol- lowing elements : 1) a collection of objects denoted by ob(C); 2) a collection of morphisms denoted by hom(C); 3) the composition of morphisms (denoted by ◦ ), that is for every a; b; c 2 ob(C) we have the binary operation hom(a; b) × hom(b; c) 7! hom(a; c) such that the following properties hold: A: The composition of morphisms is associative: 8f 2 hom(a; b); g 2 hom(b; c); h 2 hom(c; d): h ◦ (g ◦ f) = (h ◦ g) ◦ f idx: 8 x 2 ob(C), 9 idx 2 hom(x; x) s.t. 8f 2 hom(a; x) and g 2 hom(x; b): idx ◦ f = f, g ◦ idx = g In our concrete case, the Reeb graphs form a category denoted by Reeb, where the objects are finite graphs with real valued functions that are strictly monotonic on the edges, and the morphisms are functions preserv- ing maps between the underlying spaces. More precisely [3], an object of the category Reeb is a finite graph, seen as a topological space X, together 91 with a real valued function that is strictly monotonic on its edges. In what follow, we use the notation (X; f) for a such object. A morphism between (X; f) and (Y; g), is a function preserving maps ' : X ! Y , that is the following diagram is commutative: The vertices of the Reeb graph R(f) are the classes of critical points x s s such that x 2 Mc and Mc \ C[f] 6= ?. Moreover, f induces the function fe : R(f) ! R such that the following relation holds f = fe◦ π (see [1] for definition and properties about critical points). 2. The algorithm 2.1. Description of the algorithm. (1) Identify critical points; (2) Arrange critical points in a vector v[i] in the same order that they appear when tracing along the surface from absolute minimum to absolute maximum; (3) Every entry in the vector has the following form: v[i] = (Pi; type), where type can be: • a for minimum, with the special type aA for the absolute mini- mum; • b for a saddle point with the special types bi for the saddle that comes after the absolute minimum and bp the saddle that comes after a minimum (note: categ.sp). If the saddle does not come after any type of minimum, the type will be obtained by alternating the order from the last saddle that falls in the categ.sp. • c for maximum, with the special type cA for the absolute max- imum; (4) Pseudocode algorithm: Entry data: X= surface, f= real valued function Define a boolean function ver.type(P,Q) that compares the main type of two given points; Example: ver.type(a,aA)=true; ver.type (c,cA)=true, ver.type(bp, bi)= true, ver.type(a,c)=false; Define a function point(v[i]) that returns the point Pi; First result: v[i] = (Pi; type), n = lenght (v[i]); Let i = natural number; Initialize i=1; While i < n if type(v[i])=a if ver.type(point(v[i]), point(v[i+1])) is false then draw element I; 92 Adela Lupescu i=i+1; else draw element II; i=i+2; if type(v[i])=bi and type(v[i+1])=bp then draw element III; i=i+1; if type(v[i])=bp and type(v[i+1])=bi then draw element I; i=i+1; if type(v[i])=b and v[i+1]=c if ver.type(point(v[i+1]), point(v[i+2])) is false then draw element I; i=i+1; else draw element IV; i=i+2; 2.2. Examples of applying the algorithm. (1) X=surface, f(x; y; z) = z v[i] = ((P1; aA); (P2; bi); (P3; bp); (P4; bi); (P5; bp); (P6; cA)) n=6; i=1; 1 < 6 type v[1]=a (true) ver.type(point(v[1]), point(v[2])) is false (true) draw element I; i=i+1; 2 < 6 type v[2]=a (false) type(v[2])=bi and type(v[3])=bp (true) then draw element III; i=i+1; 3 < 6 type v[3]=a (false) type(v[3])=bi and type(v[4])=bp (false) type(v[3])=bp and type(v[4])=bi (true) then draw element I; i=i+1; 4 < 6 type v[4]=a (false) 93 type(v[4])=bi and type(v[5])=bp (true) then draw element III; i=i+1; 5 < 6 type v[5]=a (false) type(v[5])=bi and type(v[6])=bp (false) type(v[5])=bp and type(v[6])=bi (false) type(v[5])=bp and v[6]=c (true) ver.type(point(v[6]), point(v[7])) is false (true) then draw element I; i=i+1; 6 < 6 false draw graph; (2) X=surface, f(x; y; z) = z v[i] = ((P1; aA); (P2; a); (P3; bi); (P4; bp); (P5; bi); (P6; bp); (P7; bi); (P8; cA)) n=8; i=1; 1 < 6 type v[1]=a (true) ver.type(point(v[1]), point(v[2])) is false (false) draw element II; i=i+2; 3 < 6 type v[3]=a (false) type(v[3])=bi and type(v[4])=bp (false) type(v[3])=bp and type(v[4])=bi (true) then draw element I; i=i+1; 4 < 8 type v[4]=a (false) type(v[4])=bi and type(v[5])=bp (true) then draw element III; i=i+1; 5 < 8 type v[5]=a (false) type(v[5])=bi and type(v[6])=bp (false) type(v[5])=bp and type(v[6])=bi (true) then draw element I; i=i+1; 94 Adela Lupescu 6 < 8 type v[6]=a (false) type(v[6])=bi and type(v[7])=bp (true) then draw element III; i=i+1; 7 < 8 type v[7]=a (false) type(v[7])=bi and type(v[8])=bp (false) type(v[7])=bp and type(v[8])=bi (false) type(v[7])=b and v[8]=c (true) ver.type(point(v[8]), point(v[9])) is false (true) then draw element I; i=i+1; 8 < 8 false draw graph; References [1] Andrica, D., Critical Point Theory and Some Applications, Cluj University Press, Cluj-Napoca, 2005. [2] Bauer, U. and Ge, X. and Wang, Y., Measuring Distances between Reeb Graphs, July 2013. [3] Bauer, U. and Munch, E. and Wang,Y., Strong Equivalence of the Interleaving and Functional Distortion Metrics for Reeb Graphs, 31st International Symposium on Com- putational Geometry (SoCG 2015), LIPIcs 34 (2015), 461-475. [4] Biasotti, S. and Giorgi, D. and Spagnuolo, M. and Falcidieno, B., Reeb graphs for shape analysis and applications, Theoretical Computer Science 392 (2008), 5-22. [5] Kaluba, M. and Marzantowicz, W. and Nelson, S., On a reprezentation of the Reeb Graph as a sub-complex of manifold, Topological Methods in Nonlinear Analysis, May 2014. [6] Duke University Lectures, Computational Topology : Lecture V.4. [7] Reeb, G. , Sur les points singuliers d'une forme de Pfaff completment intgrable ou d'une fonction numerique , Comptes Rendus de l'Academie des Sciences, 222 (1946) 847-849. "BABES¸-BOLYAI" UNIVERSITY FACULTY OF MATHEMATICS AND COMPUTER SCIENCE KOGALNICEANU 1, 400084 CLUJ-NAPOCA, ROMANIA E-mail address: [email protected].