We show that there is no (1 + ε)-approximation algorithm for the problem of covering points in the plane by minimum number of fat triangles of similar size (with the minimum angle of the triangles being close to 45 degrees). Here, the available triangles are prespecified in advance. Since a constant factor approximation algorithm is known for this problem [CV07], this settles the approximabilit...

Let AA A A be a triangle with vertices at A1, A2 and A3. The process of "bisecting AA A A is defined as follows. We first locate the longest edge, A'Ai+i of AA 1A2A3 where A'+3 = Al, set D = (A' + Ai+l)/2, and then define two new triangles, AA'DAi+2 and ADAi+1Ai+2. Let Aqq be a given triangle, with smallest interior angle a > 0. Bisect AQ0 into two new triangles, Aj., í = 1, 2. Next, bisect eac...

We show that every K4-free planar graph with at most ν edge-disjoint triangles contains a set of at most 32ν edges whose removal makes the graph triangle-free. Moreover, equality is attained only when G is the edge-disjoint union of 5-wheels plus possibly some edges that are not in triangles. We also show that the same statement is true if instead of planar graphs we consider the class of graph...

We have shown previously that the precision of angle judgments depends strongly on the global stimulus configuration: discrimination thresholds for angles that form part of isosceles triangles are up to 3 times lower than for those that form part of scalene triangles [Kennedy, G. J., Orbach, H. S., & Loffler, G. (2006). Effects of global shape on angle discrimination. VisionResearch, 46(8-9), 1...

In this note we introduce a new randomized algorithm for counting triangles in graphs. We show that under mild conditions, the estimate of our algorithm is strongly concentrated around the true number of triangles. Specifically, if p ≥ max ( log n t , log n √ t ), where n, t, ∆ denote the number of vertices in G, the number of triangles in G, the maximum number of triangles an edge of G is cont...

Information on su(N) tensor product multiplicities is neatly encoded in Berenstein-Zelevinsky triangles. Here we study a generalisation of these triangles by allowing negative as well as positive integer entries. For a fixed triple product of weights, these generalised Berenstein-Zelevinsky triangles span a lattice in which one may move by adding integer linear combinations of so-called virtual...

A visibility algorithm for triangulated panoramas is proposed. The algorithm can correctly resolve the visibility without making use of any depth information. It is especially useful when depth information is not available, such as in the case of real-world photographs. Based on the optical flow information and the image intensity, the panorama is subdivided into variable-sized triangles, image...

The Laplacian-energy-like of a simple connected graph G is defined as LEL:=LEL(G)=∑_(i=1)^n√(μ_i ), Where μ_1 (G)≥μ_2 (G)≥⋯≥μ_n (G)=0 are the Laplacian eigenvalues of the graph G. Some upper and lower bounds for LEL are presented in this note. Moreover, throughout this work, some results related to lower bound of spectral radius of graph are obtained using the term of ΔG as the num...