A new proof of the flat wall theorem

We give an elementary and self-contained proof, and a numerical improvement, of a weaker form of the excluded clique minor theorem of Robertson and Seymour, the following. Let t,r >= 1 be integers, and let R = 49152t(24) (40t(2) +r). An r-wall is obtained from a 2r x r-grid by deleting every odd vertical edge in every odd row and every even vertical edge in every even row, then deleting the two resulting vertices of degree one, and finally subdividing edges arbitrarily. The vertices of degree two that existed before the subdivision are called the pegs of the r-wall. Let G be a graph with no Kt minor, and let W be an R-wall in G. We prove that there exist a set A subset of V(G) of size at most 12288t(24) and an r-subwall W' of W such that V(W') n A = 0 and W' is a flat wall in G A in the following sense. There exists a separation (X, Y) of G A such that X boolean AND Y is a subset of the vertex set of the cycle C' that bounds the outer face of W', V(W') subset of Y, every peg of W' belongs to X and the graph G[Y] can almost be drawn in the unit disk with the vertices X n Y drawn on the boundary of the disk in the order determined by C'. Here almost means that the assertion holds after repeatedly removing parts of the graph separated from X n Y by a cutset Z of size at most three, and adding all edges with both ends in Z. Our proof gives rise to an algorithm that runs in polynomial time even when r and t are part of the input instance. The proof is self-contained in the sense that it uses only results whose proofs can be found in textbooks. (C) 2017 The Authors. Published by Elsevier Inc

Structural analysis of three space crane articulated-truss joint concepts

Three space crane articulated truss joint concepts are studied to evaluate their static structural performance over a range of geometric design parameters. Emphasis is placed on maintaining the four longeron reference truss performance across the joint while allowing large angle articulation. A maximum positive articulation angle and the actuator length ratio required to reach the angle are computed for each concept as the design parameters are varied. Configurations with a maximum articulation angle less than 120 degrees or actuators requiring a length ratio over two are not considered. Tip rotation and lateral deflection of a truss beam with an articulated truss joint at the midspan are used to select a point design for each concept. Deflections for one point design are up to 40 percent higher than for the other two designs. Dynamic performance of the three point design is computed as a function of joint articulation angle. The two lowest frequencies of each point design are relatively insensitive to large variations in joint articulation angle. One point design has a higher maximum tip velocity for the emergency stop than the other designs