Constructing of OE-Postman Path for a Planar Graph

Abstract

The model of cutting plan can be presented as a planar graph for automated system of sheet material cutting process preparation. The aim of such modelling is a denition of the shortest path of a cutter having no parts requiring any additional cuttings. The paper is devoted to a problem of chines postman path constructing for a planar graph representing a cutting plan. This path has a restriction of ordered enclosing (i.e. cycle of passed edges does not contain inside not passed ones). The path satisfying this restriction is also called OE-path. This kind of restriction means the lack of additional cuttings of details. The recursive algorithm for constructing of this type of paths is considered in the paper. It is proved that this algorithm has a polynomial complexity. The developed software allows to solve the problem for an arbitrary planar graph. The software is tested for the typical cases of planar graphs.

The estimation of the number of OE-chains and realizable OE-routes for cutting plans with combined contours

“…OE-Cycle) [19], [21] O(|E| • log 2 |E|) OE-Postman Route (alg. CPP OE) [17] O(|E| • |V |) OE-Router [1] O(|E| • log 2 |E|) Optimal OE-Router [1] O(|V | 2 ) MultiComponent (OE-Cover for Disconnected graph [18]) O(|E| • log 2 |E|) DoubleBridging (OE-Cover for Disconnected graph [18])…”

“…A number of papers [15,16,17] discuss constructing the efficient algorithms for cutting plans in which combining the contours is allowed. To solve this problem the authors of [15] use graph theory, and the algorithm considered in this paper allows to construct additional edges between odd degree vertices.…”

“…This algorithm works only for plane graphs admitting completion to a plane Euler graph. In [17] the author considers the same problem as in [15] but formalized it in details giving the definition of a route with restrictions for a plane graph (this type of routes is called OE-routes) and not only the algorithm for solving the problem for Eulerian graph but also for solving Chinese Postman Problem (CPP). Thus, algorithm for obtaining the solution of this problem is given for a plane connected graph.…”

“…We define the following functions for each edge E ∈ E(G) to represent the image of cutting plan as a plane graph G = (V, E) [17]:…”

Special Type Routing Problems in Plane Graphs

“…Add the edges of the obtained spanning tree to graph G: E( G := E( G) ∪ E(T(T )), B := B ∪ E(T(T )). 8: end for 9: end Plane graph G obtained by algorithm Bridging contains bridges, hence it is possible to apply only algorithm CPP_OE [30] constructing the Chinese postman OE-route for plane graph [30]. Note that both the OECover algorithm and the M-Cover algorithm require no bridges in the graph.…”