Chie Nara

The maximal number of induced complete bipartite graphs

Abstract The aim of this paper is to determine the maximal number of induced K(t, t) subgraphs in graphs of given order and in graphs of given size.

Affine equivalent classes of parallelohedra

In the paper the affine equivalence relation in the set of parallelohedra is studied. One proves the uniqueness theorem for a wide class of d-dimensional parallelohedra. From here it follows that for every d (≥2) the space of affine equivalent classes of d-dimensional primitive parallelohedra has dimension d(d+1)/2−1.

Continuous flattening of platonic polyhedra

We prove that each Platonic polyhedron P can be folded into a flat multilayered face of P by a continuous folding process for polyhedra.

Continuous Flattening of Convex Polyhedra

A flat folding of a polyhedron is a folding by creases into a multilayered planar shape. It is an open problem of E. Demaine et al., that every flat folded state of a polyhedron can be reached by a continuous folding process. Here we prove that every convex polyhedron possesses infinitely many continuous flat folding processes. Moreover, we give a sufficient condition under which every flat folded state of a convex polyhedron can be reached by a continuous folding process.

Continuously Flattening Polyhedra Using Straight Skeletons

We prove that a surprisingly simple algorithm folds the surface of every convex polyhedron, in any dimension, into a flat folding by a continuous motion, while preserving intrinsic distances and avoiding crossings. The flattening respects the straight-skeleton gluing, meaning that points of the polyhedron touched by a common ball inside the polyhedron come into contact in the flat folding, which answers an open question in the book Geometric Folding Algorithms. The primary creases in our folding process can be found in quadratic time, though necessarily, creases must roll continuously, and we show that the full crease pattern can be exponential in size. We show that our method solves the fold-and-cut problem for convex polyhedra in any dimension. As an additional application, we show how a limiting form of our algorithm gives a general design technique for flat origami tessellations, for any spiderweb (planar graph with all-positive equilibrium stress).

Refold Rigidity of Convex Polyhedra

Abstract We show that every convex polyhedron may be unfolded to one planar piece, and then refolded to a different convex polyhedron. If the unfolding is restricted to cut only edges of the polyhedron, we identify several polyhedra that are “edge-refold rigid” in the sense that each of their unfoldings may only fold back to the original. For example, each of the 43,380 edge unfoldings of a dodecahedron may only fold back to the dodecahedron, and we establish that 11 of the 13 Archimedean solids are also edge-refold rigid. We begin the exploration of which classes of polyhedra are and are not edge-refold rigid, demonstrating infinite rigid classes through perturbations, and identifying one infinite nonrigid class: tetrahedra.

Maximal Number of Edges in Geometric Graphs without Convex Polygons

A geometric graph G is a graph whose vertex set is a set P n of n points on the plane in general position, and whose edges are straight line segments (which may cross) joining pairs of vertices of G. We say that G contains a convex r-gon if its vertex and edge sets contain, respectively, the vertices and edges of a convex polygon with r vertices. In this paper we study the following problem: Which is the largest number of edges that a geometric graph with n vertices may have in such a way that it does not contain a convex r-gon? We give sharp bounds for this problem. We also give some bounds for the following problem: Given a point set, how many edges can a geometric graph with vertex set P n have such that it does not contain a convex r-gon?

Enumeration of unlabelled graphs with specified degree parities

Abstract This paper gives a generating function for unlabelled graphs of order n . The coefficient of each monomial in this function shows the number of unlabelled graphs with given size and the number of odd vertices. Furthermore, the numerical examples are given for 1⩽ n ⩽9.

Continuous flattening of orthogonal polyhedra

Can we flatten the surface of any 3-dimensional polyhedron P without cutting or stretching? Such continuous flat folding motions are known when P is convex, but the question remains open for nonconvex polyhedra. In this paper, we give a continuous flat folding motion when the polyhedron P is an orthogonal polyhedron, i.e., when every face is orthogonal to a coordinate axis (x, y, or z). More generally, we demonstrate a continuous flat folding motion for any polyhedron whose faces are orthogonal to the z axis or the xy plane.

Transformability and Reversibility of Unfoldings of Doubly-Covered Polyhedra

Let \(W\) and \(X\) be convex polyhedra in the 3-dimensional Euclidean space. If \(W\) is dissected into a finite number of pieces which can be rearranged to form \(X\) with hinges (which compose a dissection tree), \(W\) is called transformable to \(X\), and if the surface of \(W\) is transformed to the interior of \(X\) except some edges of pieces, \(W\) is called reversible to \(X\). Let \(P\) be a reflective space-filler in the 3-space and let \(P^m\) be a mirror image of \(P\). In this paper, we show that any convex unfolding \(W\) of the doubly covered polyhedron \(d(P)\) of \(P\) is transformable to any convex unfolding \(X\) of the doubly covered polyhedron \(d(P^m)\) of \(P^m\), where we assume that \(W\) (resp. \(X\)) includes \(P\) (resp. \(P^m\)) as a subset. Moreover if \(W\) is dissected into \(n\) non-empty pieces (where \(n\) is the number of faces of \(P\)), \(W\) is reversible to \(X\).