We develop a theory of minors for alternating dimaps --- orientably embedded digraphs where, at each vertex, the incident edges (taken in the order given by the embedding) are directed alternately into, and out of, the vertex. We show that they are related by the triality relation of Tutte. They do not commute in general, though do in many circumstances, and we characterise the situations where they do. The relationship with triality is reminiscent of similar relationships for binary functions, due to the author, so we characterise those alternating dimaps which correspond to binary functions. We give a characterisation of alternating dimaps of at most a given genus, using a finite set of excluded minors. We also use the minor operations to define simple Tutte invariants for alternating dimaps and characterise them. We establish a connection with the Tutte polynomial, and pose the problem of characterising universal Tutte-like invariants for alternating dimaps based on these minor operations.
It is interesting to note that this stream of research, first seen in Tutte's 1948 paper @cite_16 , can be traced back to the same source that eventually gave rise to Tutte's work on minor operations and his eponymous polynomial. Historically, the source of both streams was the famous paper on squaring the square'' @cite_10 . The 1948 paper extended the theory to triangulating the triangle'' (where all triangles are equilateral) and introduced triality, among other things. However, this stream has not previously seen the development of minor operations or Tutte-like invariants for alternating dimaps.
@cite_16 @cite_10
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Abstract We study the problem of reconstructing a multivariate trigonometric polynomial having only few non-zero coefficients from few random samples. Inspired by recent work of Candes, Romberg and Tao we propose to recover the polynomial by Basis Pursuit, i.e., by l 1 -minimization. In contrast to their work, where the sampling points are restricted to a grid, we model the random sampling points by a continuous uniform distribution on the cube, i.e., we allow them to have arbitrary position. Numerical experiments show that with high probability the trigonometric polynomial can be recovered exactly provided the number N of samples is high enough compared to the “sparsity”—the number of non-vanishing coefficients. However, N can be chosen small compared to the assumed maximal degree of the trigonometric polynomial. We present two theorems that explain this observation. One of them provides the analogue of the result of Candes, Romberg and Tao. The other one is a result toward an average case analysis and, unexpectedly connects to an interesting combinatorial problem concerning set partitions, which seemingly has not yet been considered before. Although our proofs follow ideas of they are simpler. We give a simple combinatorial algorithm that computes a piecewise-linear approximation of a smooth surface from a finite set of sample points. The algorithm uses Voronoi vertices to remove triangles from the Delaunay triangulation. We prove the algorithm correct by showing that for densely sampled surfaces, where density depends on a local feature size function, the output is topologically valid and convergent (both pointwise and in surface normals) to the original surface. We briefly describe an implementation of the algorithm and show example outputs. This paper studies Newton-type methods for minimization of partly smooth convex functions. Sequential Newton methods are provided using local parameterizations obtained from **-Lagrangian theory and from Riemannian geometry. The Hessian based on the **-Lagrangian depends on the selection of a dual parameter g; by revealing the connection to Riemannian geometry, a natural choice of g emerges for which the two Newton directions coincide. This choice of g is also shown to be related to the least-squares multiplier estimate from a sequential quadratic programming (SQP) approach, and with this multiplier, SQP gives the same search direction as the Newton methods. We study Hanani-Tutte style theorems for various notions of planarity, including partially embedded planarity, and simultaneous planarity. This approach brings together the combinatorial, computational and algebraic aspects of planarity notions and may serve as a uniform foundation for planarity, as suggested in the writings of Tutte and Wu.