A graph is fan-crossing free if it has a drawing in the plane so that each edge crossed by independent edges, crossing edges have distinct vertices. On other hand, common vertex, they form fan. Both are prominent examples for beyond-planar graphs. Further well-known classes k-planar, k-gap-planar, quasi-planar, and right angle We use subdivision, node-to-circle expansion path-addition operation...

We introduce a new class of subdivision surfaces which generalize uniform tensor product B-spline surfaces of any bi-degree to meshes of arbitrary topology. Surprisingly , this can be done using subdivision rules that involve direct neighbors only. Consequently, our schemes are very easy to implement, regardless of degree. The famous Catmull-Clark scheme is a special case. Similarly we show tha...

A new technique for the reconstruction of a smooth surface from a set of 3D data points is presented. The reconstructed surface is represented by an everywhere Ccontinuous subdivision surface which interpolates all the given data points. The new technique consists of two major steps. First, an ef£cient surface reconstruction method is applied to produce a polyhedral approximation to the given d...

The in-ation GI of a graph G is the line graph of the subdivision of G. If G is a complete graph the equality ir(GI) = (GI) was proved by Favaron in 1998. We conjectured that the equality holds when G is any graph of radius 1. But it turned out that it is not true. Moreover, we proved that for the class of radius 1 graphs there does not exist a better upper bound for the relation (GI)=ir(GI) th...

Partitioning a graph into three pieces, with two of them large and connected, and the third a small “separator” set, is useful for improving the performance of a number of combinatorial algorithms. This is done using the second eigenvector of a matrix defined solely in terms of the incidence matrix, called the graph Laplacian. For sparse graphs, the eigenvector can be efficiently computed using...

We prove a decomposition result for locally finite graphs which can be used to extend results on edge-connectivity from finite to infinite graphs. It implies that every 4k-edge-connected graph G contains an immersion of some finite 2k-edge-connected Eulerian graph containing any prescribed vertex set (while planar graphs show that G need not contain a subdivision of a simple finite graph of lar...

Quad/triangle subdivision unifies triangular and quadrilateral subdivision schemes. Despite its interest, this scheme is rarely used in the multiresolution edition framework, mainly because of the lack of a sufficiently general data structure that allows a simple and efficient implementation of multiresolution meshes built with it. We show in this paper how multiresolution half-edges, defined a...