Computing an invariant of a graph such as treewidth and pathwidth is one of the fundamental problems in graph algorithms. In general, determining the pathwidth of a graph is NP-hard. In this paper, we propose several reduction methods for decreasing the instance size without changing the pathwidth, and implemented the methods together with an exact algorithm for computing pathwidth of graphs. Our experimental results show that the number of vertices in all chemical graphs in NCI database decreases by our reduction methods by 53.81% in average.

A single-mode fiber employing a trench index profile for indoor wiring is proposed to realize low-bending loss and low-splice loss simultaneously. The designs and the advantages of the fiber over fibers with conventional step index profiles are described. The characteristics of manufactured fibers with trench index profiles are also described. One of the manufactured fibers realizes a bending loss of 0.018 dB/turn at 1550 nm for a bending radius of 7.5 mm, and a splice loss of 0.19 dB at 1550 nm in mechanical splicing to a conventional single mode fiber simultaneously. Other one of the fibers realizes a bending loss of 0.011 dB/turn at 1550 nm for a bending radius of 5 mm, and the splice loss is 0.37 dB at 1550 nm simultaneously. A total loss of a fiber employing the trench index is small and stable against the fluctuation of MFD.