Resource-efficient technologies for cutting sheet materials allow for the contours of cut-off details to be overlapped. This includes the Intermittent Cutting Problem and the Endpoint Cutting Problem. This paper reviews mathematical models of such resource-efficient cutting processes and suggests algorithms for defining the cutter route under technological constraints. As soon as a cutting plan is given, optimisation with shortening the total length of idle passes. No information of the detail shape is required to define the sequence of detail cutting. This is why all curves without self-intersections and contiguities that are a constituent part of detail boundaries may be interpreted as edges of the plane graph G, and all points of intersection and contiguity may be interpreted as vertices of the graph G. Up to homeomorphism, plane graph G can be represented by a list of edges with incident vertices and faces . This allows the restrictions of the planed cutter trajectory to be formalised in terms of graph theory. A series of algorithms for constructing the permitted route in the plane graph G as an image of the cutting plan is suggested. The constructed route for the graph G can be interpreted as a tool trajectory for the cutting plan which is the inverse image of graph G.

Resource-conscious technologies for cutting sheet material include the ICP and ECP technologies that allow for aligning fragments of the contours of cutouts. In this work, we show the mathematical model for the problem of cutting out parts with these technologies and algorithms for finding cutting tool routes that satisfy technological constraints. We give a solution for the problem of representing a cutting plan as a plane graph G = (V,F,E), which is a homeomorphic image of the cutting plan. This has let us formalize technological constraints on the trajectory of cutting the parts according to the cutting plan and propose a series of algorithms for constructing a route in the graph G = (V,F,E), which is an image of an admissible trajectory. Using known coordinates of the preimages of vertices of graph G = (V,F,E) and the locations of fragments of the cutting plan that are preimages of edges of graph G = (V,F,E), the resulting route in the graph G = (V,E) can be interpreted as the cutting tool’s trajectory. The proposed algorithms for finding routes in a connected graph G have polynomial computational complexity. To find the optimal route in an unconnected graph G, we need to solve, for every dividing face f of graph G, a travelling salesman problem on the set of faces incident to f.