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Graph-encoded maps

Abstract An edge-colored graph model for the study of finite graphs embedded in surfaces (maps) is presented. This model brings to light the symmetry among three kinds of dualities on maps: (a) the usual duality that interchanges vertices and faces; (b) a duality that interchanges vertices and certain closed walks called zigzags; (c) a duality that interchanges faces and zigzags. Moreover, the orientability of the surface of a map is shown to be equivalent to the bipartiteness of the colored graph.

An L-approach for packing (ℓ, w)-rectangles into rectangular and L-shaped pieces

This paper presents an approach using a recursive algorithm for packing (ℓ, w)-rectangles into larger rectangular and L-shaped pieces. Such a problem has actual applications for non-guillotine cutting and pallet/container loading. Our motivation for developing the L-approach is based on the fact that it can solve difficult pallet loading instances. Indeed, it is able to solve all testing problems (more than 20 000 representatives of infinite equivalence classes of the literature), including the 18 hard instances unresolved by other heuristics. We conjecture that the L-approach always finds optimum packings of (ℓ, w)-rectangles into rectangular pieces. Moreover, the approach may also be useful when dealing with cutting and packing problems involving L-shaped pieces.

An n-tet graph approach for non-guillotine packings of n-dimensional boxes into an n-container

Abstract In this paper we propose a simple recursive uniform algorithm for the problem of packing n -dimensional boxes into an n -container. We are particularly concerned about the special case n =3 where the boxes can be packed in a given subset of their six possible positionings. Our method studies symmetries in the packings by the use of an ordered set of three directed graphs with the same edges (a 3- tet or triad ) and induced smaller structures of the same kind named minors . With the method, degeneracy and symmetry issues, which curtail the implicit enumeration to practically acceptable running times, become transparent. In order to illustrate the performance of the algorithm, computational results from solving randomly generated 3-D examples are presented and compared with the ones of a layers and knapsack approach. The present study has real world applications for the problems of pallet and container loading.

The Gauss code problem off the plane

The classical Gauss code problem asks to characterize which cyclic sequences arise as the vertex sequence of the ‘straight-ahead path of a 4-regular graph embedded in the plane This problem is generalized to certain 4 regular graphs in arbitrary surfacesA characterization is given for the existence of a 4-regular graph in a specified surface yielding the specified sequence This characterization is obtained using a generalization of Shanks left-right paths

A minimax theorem on circuits in projective graphs

Abstract A minimax theorem is proved. The theorem concerns packing non-separating circuits in eulerian graphs embedded in the projective plane. The proof includes a polynomial algorithm which produces a collection of edge-disjoint, non-separating circuits of the same cardinality as a transversal of such circuits.

Combinatorics of orientation reversing polygons

We give two combinatorial characterizations of orientation reversing polygons in graphs embedded on surfaces (r-polygons) and use the notion of skew embedding introduced in [7] to characterize “parity” embeddings: an embedding has its odd polygons coinciding with itsr-polygons if and only if the skew embedding is in an orientable surface. The concept of “imbalance”, central for the proof, does not seem to appear explicitly before in the literature. Possible algorithmic implications of the parity embedding theorem are briefly discussed.

BLOBS AND FLIPS ON GEMS

In this paper we prove that two n-gems induce the same manifold if and only if they are linked by a finite sequence of gem moves. A gem move is either a blob move, consisting in the creation or cancellation of an n-dipole, or a clean flip, which is a switch of a pair of edges of the same color that thickens an h-dipole, 1 ≤ h ≤ n - 1, or the inverse operation, which slims an h-dipole, 2 ≤ h ≤ n. Moreover we prove that we can reorder the gem moves, so that all the blob creations precede all clean flips which then precede all the blob cancellations. This reordering is of interest because it is an easy matter to decide whether two gems are linked by a finite sequence of clean flips. As a consequence, if a bound for the number of blob creations is established, then there exists a deterministic finite algorithm to decide whether two gems induce the same manifold or not.

Phorma: perfectly hashable order restricted multidimensional arrays

In this paper we propose a simple and efficient data structure yielding a perfect hashing of quite general arrays. The data structure is named phorma, which is an acronym for perfectly hashable order restricted multidimensional array.

Stereoisomerism in Lanthanide Complexes: Enumeration, Chirality, Identification, Random Coordination Ratios

The concept of random coordination ratios, RCRs, is advanced for lanthanide complexes. RCRs describe the relative probabilities of occurrence of subsets of stereoisomers of same-symmetry point groups in the limiting situation when energetic effects are equivalent. We then introduce a method to uniquely identify the stereoisomer of the coordination polyhedron of a given crystallographic structure and introduce a notation that fully characterizes its stereochemistry in an unambiguous manner, from which absolute configuration naturally follows. De facto, the coordination chirality in lanthanide complexes is a frequently overlooked property, even though these compounds often exhibit, when luminescent, high dissymmetry factors. With our methodology, we even managed to recognize a known dilanthanide complex as a meso compound, with both metal ions functioning as stereogenic centers. To achieve these results, we enumerate all possible stereoisomers of lanthanide complexes with coordination numbers from 4 to 9 for all combinations of monodentate, symmetric and asymmetric bidentate ligands, and for several shapes of coordination polyhedra. We confirmed the number of stereoisomers for each case by means of Pólyas theorem. We further classified all stereoisomers according to their symmetry point groups and generated their Cartesian coordinates. This collection of all coordination polyhedra stereoisomer geometries, which is made available in the Supporting Information , can also be used to easily build starting-point geometries for theoretical calculations of metal complexes.

Combinatorial Dehn-Lickorish Twists and Framed Link Presentations of 3-Manifolds Revisited

From a pseudo-triangulation with n tetrahedra T of an arbitrary closed orientable connected 3-manifold (for short, a 3D-space) M3, we present a gem J′, inducing 𝕊3, with the following characteristics: (a) its number of vertices is O(n); (b) it has a set of p pairwise disjoint couples of vertices {ui, vi}, each named a twistor; (c) in the dual (J′)⋆ of J′ a twistor becomes a pair of tetrahedra with an opposite pair of edges in common, and it is named a hinge; (d) in any embedding of (J′)⋆ ⊂ 𝕊3, the ∊-neighborhood of each hinge is a solid torus; (e) these p solid tori are pairwise disjoint; (f) each twistor contains the precise description on how to perform a specific surgery based in a Denh–Lickorish twist on the solid torus corresponding to it; (g) performing all these p surgeries (at the level of the dual gems) we produce a gem G′ with |G′| = M3; (h) in G′ each such surgery is accomplished by the interchange of a pair of neighbors in each pair of vertices: in particular, |V(G′) = |V(J′)|. This is a new p...