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M/ is iteratively constructed through a breadth-first traversal of the 1-skeleton of the simplicial mesh ˙ (formed by its vertices and edges). For each vertex p, its adjacent, and already processed, vertices in the mesh are examined. m/. If they belong to two or more ascending regions, then p is marked as a watershed point. Vertices that are not connected to any previously processed vertex are new minima and they start a new ascending region. Each maximal simplex (an n-simplex if we consider an n-dimensional simplicial mesh) is assigned to an ascending region based on the labels of its vertices.

If p is a 2-saddle, its lower link is a deformation retract of an annulus. Vertex p starts a new separating 2-cell. A cycle that encircles the whole lower link of p is created. All triangles and edges connecting p to this cycle are marked. They form the initial separating 2-cell associated with p. Other non-labeled simplexes in the star of p are labeled by copying from neighbors. • If p is a minimum, its lower link is empty, and p is marked. The descending 3-cells of maxima produced by the algorithm in [19] are topological cells.

The face 9741, which is a coface of 94 with exactly one unpaired face, is inserted in PQone and edge 94 is removed from PQzero (Fig. 14c). Face 9741 is taken from PQone and paired with edge 97, which is removed from PQzero. Face 9765 is inserted in PQone and successively removed to be paired with edge 95 (Fig. 14d, e). Face 9852 is removed from PQone and declared critical, as it has no unpaired faces (Fig. 14f). ˇ L. Comi´ c et al. 24 In the 3D case, the algorithm in [38] does not create spurious critical cells.

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Algebraic Graph Theory by N. Biggs

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