Guaranteed upper bounds on the length of a shortest cycle through k ≤ 5 prescribed vertices of a polyhedral graph or plane triangulation are proved.

In this note we prove an explicit formula for the lower semicontinuous envelope of some functionals defined on real polyhedral chains. More precisely, denoting by H : R → [0,∞) an even, subadditive, and lower semicontinuous function with H(0) = 0, and by ΦH the functional induced by H on polyhedral m-chains, namely

We analyze the asymptotic convergence of all infinite products of matrices taken in a given finite set by looking only at finite or periodic products. It is known that when the matrices of the set have a common nonincreasing polyhedral norm, all infinite products converge to zero if and only if all infinite periodic products with periods smaller than a certain value converge to zero. Moreover, ...

A binary three-dimensional (3D) image I is well-composed if the boundary surface of its continuous analog is a 2D manifold. In this paper, we present a method to locally “repair” the cubical complex Q(I) (embedded in R3) associated to I to obtain a polyhedral complex P (I) homotopy equivalent to Q(I) such that the boundary surface of P (I) is a 2D manifold (and, hence, P (I) is a well-composed ...

We begin by defining and studying tightness and the two-piece property for smooth and polyhedral surfaces in three-dimensional space. These results are then generalized to surfaces with boundary and with singularities, and to surfaces in higher dimensions. Later sections deal with generalizations to the case of smooth and polyhedral submanifolds of higher dimension and codimension, in particula...

The ballast of the railway track is constantly changing due to dynamic forces of train traffic that results in the crushing of the rocks. The feasibility to simulate each particle makes the discrete element method (DEM) suitable for the task. Different DEM particle models are introduced in our paper and the adequate one was chosen. This new method is based on crushable convex polyhedral element...

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, ...

Polyhedral embeddings of cubic graphs by means of certain operations are studied. It is proved that some known families of snarks have no (orientable) polyhedral embeddings. This result supports a conjecture of Grünbaum that no snark admits an orientable polyhedral embedding. This conjecture is verified for all snarks having up to 30 vertices using computer. On the other hand, for every nonorie...

This paper presents deployment strategies for a team of multiple mobile robots with line-of-sight visibility in 1.5D and 2.5D terrain environments. Our objective is to distributively achieve full visibility of a polyhedral environment. In the 1.5D polyhedral terrain, we achieve this by determining a set of locations that the robots can distributively occupy. In the 2.5D polyhedral terrain, we a...

In the last decade our capability of solving integer programming problems has increased dramatically due to the effectiveness of cutting plane methods based on polyhedral investigations. Polyhedral cutting planes have become central features in optimization software packages for integer programming. Here we present some of the important polyhedral methods used in discrete optimization. We discu...