Graphs of polyhedra; polyhedra as graphs

Graphs of polyhedra; polyhedra as graphs

Relations between graph theory and polyhedra are presented in two contexts. In the first, the symbiotic dependence between 3-connected planar graphs and convex polyhedra is described in detail. In the second, a theory of nonconvex polyhedra is based on a graph-theoretic foundation. This approach eliminates the vagueness and inconsistency that pervade much of the literature dealing with polyhedr...

The number of corner polyhedra graphs

Corner polyhedra were introduced by Eppstein and Mumford (2014) as the set of simply connected 3D polyhedra such that all vertices have non negative integer coordinates, edges are parallel to the coordinate axes and all vertices but one can be seen from infinity in the direction (1, 1, 1). These authors gave a remarkable characterization of the set of corner polyhedra graphs, that is graphs tha...

Algorithms for Drawing Polyhedra from 3-Connected Planar Graphs

Two algorithms for producing polyhedral representations for 3-connected planar graphs are discussed in the paper. One of them uses Tutte’s drawing algorithm [11] to produce 2D drawing. Then the drawing is lifted into 3D space obtaining a polyhedral embedding. The other is simple algorithm by G. Hart [4] for drawing canonical polyhedral representations. Some alternative aspects (physical model, ...

Learning Preferences Graphs by Soft Projections onto Polyhedra

We discuss the problem of learning to predict the order of nodes in a graph from a real valued feedback associated with each node. This setting includes as special cases binary classification, multiclass categorization, and multilabel ordering. In the preference graph problem the nodes are labels and edges express preferences over labels. We approach the problem by defining a loss function for ...

Are Your Polyhedra the Same as My Polyhedra?