A-trails and their application to industrial process

This paper is devoted to extension of the mathematical theory of A-trails and OE-trails for planar graphs. These types of trails imply constructing such trails avoiding intersection of not-passed edges with passed cycles and ensuring absence of intersected transitions at each graph vertex. In terms of cutting process, a planar graph is to be the cutting plan; the process of constructing such a trail means constructing a cutter path when a part cut off a sheet does not require additional cuts, and A-trail constructing means absence of cuts intersections. The polynomial algorithm for constructing the AOE-trail for 4-regular graph is considered in this paper. Such type of graph can be obtained after abstracting a cutting plan composed of rectangular parts and adding all idle passes.