The flip graph of triangulations has as vertices all triangulations of a convex n-gon, and an edge between any two triangulations that differ in exactly one edge. An r-rainbow cycle in this graph is a cycle in which every inner edge of the triangulation appears exactly r times. This notion of a rainbow cycle extends in a natural way to other flip graphs. In this paper we investigate the existence of r-rainbow cycles for three different flip graphs on classes of geometric objects: the aforementioned flip graph of triangulations of a convex n-gon, the flip graph of plane trees on an arbitrary set of n points, and the flip graph of non-crossing perfect matchings on a set of n points in convex position. In addition, we consider two flip graphs on classes of non-geometric objects: the flip graph of permutations of {1, 2, . . . , n} and the flip graph of k-element subsets of {1, 2, . . . , n}. In each of the five settings, we prove the existence and nonexistence of rainbow cycles for different values of r, n and k.
We study straight-line drawings of planar graphs where every inner face has a prescribed area. A plane graph is area-universal if for every assignment of positive real numbers to the inner faces, there exists a straight-line drawing in which the area of each inner face equals to the assigned number. The only non-area-universal graphs known so far are those containing the octahedron graph as a subgraph.We give a simple counting argument that allows us to prove non-area-universality for a large class of triangulations, namely Eulerian triangulations. Adding some geometric arguments, the concept can be used to show non-area-universality for other graphs such as the icosahedron graph.Relaxing the straight-line property by allowing the edges to bend, we show that one bend per edge is sufficient to realize any face area assignment of every plane graph, i.e., the 1-subdivision of every plane graph is area-universal. For the subclass of plane bipartite graphs, it even suffices that half of the edges have a bend.not area-universal area-universal * A preliminary version appeared in the proceedings of WG 2016 [18].
We study the problem of convexifying drawings of planar graphs. Given any planar straightline drawing of an internally 3-connected graph, we show how to morph the drawing to one with strictly convex faces while maintaining planarity at all times. Our morph is convexity-increasing, meaning that once an angle is convex, it remains convex. We give an efficient algorithm that constructs such a morph as a composition of a linear number of steps where each step either moves vertices along horizontal lines or moves vertices along vertical lines. Moreover, we show that a linear number of steps is worst-case optimal. To obtain our result, we use a well-known technique by Hong and Nagamochi for finding redrawings with convex faces while preserving y-coordinates. Using a variant of Tutte's graph drawing algorithm, we obtain a new proof of Hong and Nagamochi's result which comes with a better running time. This is of independent interest, as Hong and Nagamochi's technique serves as a building block in existing morphing algorithms.
Given a neural network, training data, and a threshold, it was known that it is NP-hard to find weights for the neural network such that the total error is below the threshold. We determine the algorithmic complexity of this fundamental problem precisely, by showing that it is ∃R-complete. This means that the problem is equivalent, up to polynomial time reductions, to deciding whether a system of polynomial equations and inequalities with integer coefficients and real unknowns has a solution. If, as widely expected, ∃R is strictly larger than NP, our work implies that the problem of training neural networks is not even in NP.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.