Whitney's theorem states that 3-connected planar graphs admit essentially unique embeddings in the plane. We generalize this result to embeddings of graphs in arbitrary surfaces by showing that there ...is a function ξ:N0→N0 such that every 3-connected graph admits at most ξ(g) combinatorially distinct embeddings of face-width ⩾3 into surfaces whose Euler genus is at most g.
We show that every 5-connected graph admitting an embedding into the projective plane with face-width at least 3 contains
K
6 as a minor. Further we find a simple proof that every 5-connected planar ...graph contracts to icosahedron. We adapt the proof and find a simple face-width type condition such that every 5-connected graph embedded in an arbitrary surface satisfying the condition contracts to icosahedron.
Let G be an unweighted graph of complexity n cellularly embedded in a surface (orientable or not) of genus g. We describe improved algorithms to compute (the length of) a shortest non-contractible ...and a shortest non-separating cycle of G.
If k is an integer, we can compute such a non-trivial cycle with length at most k in O(gnk) time, or correctly report that no such cycle exists. In particular, on a fixed surface, we can test in linear time whether the edge-width or face-width of a graph is bounded from above by a constant. This also implies an output-sensitive algorithm to compute a shortest non-trivial cycle that runs in O(gnk) time, where k is the length of the cycle.
This paper presents a study on the effect of extra face width on root and contact stresses of spur and helical gear pairs. The root and contact stresses of spur and helical gear pairs, in which extra ...face width is not in contact, were calculated using Kubo and Umezawa's method. The validity of the calculation method was confirmed by comparing the calculated root stresses of spur gear pairs with extra face width with measured ones. The effect of the extra face width on the root and contact stresses of spur and helical gear pairs were determined to a considerable extent. The changes of maximum root and contact stresses of the gear pairs with extra face width were found to be fairly small.
This paper presents a study on the root and contact stresses of WN (Haseg SymMarC) gears. The validity of the method for calculating root and contact stresses of Haseg SymMarC gear teeth in mesh was ...confirmed by comparing the calculated root and contact stresses of Haseg SymMarC gear teeth with measured stresses. The root and contact stresses of Haseg SymMarC gears with various dimensions were calculated, and the effects of center distance, helix angle, face width, and number of teeth on the maximum root and contact stresses of Haseg SymMarC gear teeth in mesh were determined to a considerable extent. Furthermore, the root and contact stresses of Haseg SymMarC gear teeth were compared with those of SymMarC and involute gear teeth. The change of maximum root and contact stress of Haseg SymMarC gear with center distance is much smaller than that of SymMarC gear, and the maximum root and contact stresses of Haseg SymMarC gear were found to be much larger than that of SymMarC and involute gear teeth in the case of the normal center distance.
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