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Graphs
Published in B. V. Senthil Kumar, Hemen Dutta, Discrete Mathematical Structures, 2019
B. V. Senthil Kumar, Hemen Dutta
If there is an Eulerian path, as we follow, each vertex except the starting and ending vertices must have equal in-degree and out-degree, since whenever we come to a vertex along an edge, we leave it along another edge. The starting vertex must have out-degree 1 larger than its in-degree, since we use one edge leading out of this vertex and whenever we visit it again, we use one edge leading into it and one leaving it.
Graph Theory
Published in Rowan Garnier, John Taylor, Discrete Mathematics, 2020
An Eulerian path seeks to travel along every edge of the graph (once) and return to the starting position. An analogous problem is whether we can visit every vertex once, without travelling along any edge more than once, and return to the starting position. This problem was considered by Hamilton† (although he was not the first to do so) and his name is now associated with these paths.
Optimal order picker routing in a conventional warehouse with two blocks and arbitrary starting and ending points of a tour
Published in International Journal of Production Research, 2020
Makusee Masae, Christoph H. Glock, Panupong Vichitkunakorn
Conversely, assume that the degrees of and in the subgraph are odd, while the degrees of the other vertices are even. From the Eulerian path theorem, there exists a tour that starts at , ends at , and traverses every edge exactly once. Since the degrees of and are positive, the tour is an order picking tour. Hence, the subgraph is a tour subgraph.
Mathematical models and routing algorithms for economical cutting tool paths
Published in International Journal of Production Research, 2018
T.A. Makarovskikh, A.V. Panyukov, E.A. Savitskiy
The problems of reducing consumption and combining cuts can be solved during details placement. Dewil, Vansteenwegen, and Cattrysse (2016) mention that to solve the ECP problem Manber and Israni (1984) suggest an algorithm that finds the cutting path with minimum number of pierce points by adding bridges between odd degree vertices. To solve this problem, the authors of this paper present an algorithm that finds a Eulerian path satisfying the precedence constraint (a part cut off a sheet does not require additional cuts). Nevertheless, the algorithm by Manber and Israni (1984) works only for a plane graph which can be completed to a plane Eulerian graph.
An Online Course Content for Undergraduate Students on Full-Custom Design of a Digital VLSI Circuit Using Open-Source Software
Published in IETE Journal of Education, 2023
A good gate sequence can result in a compact stick diagram making a compact layout design of a CMOS circuit cell. The use of Euler’s graph to find the gate pattern is given in Ref. [9]. Euler’s graph needs to be traced throughout the circuit to find the best gate sequence connecting every edge, this is called the Eulerian path, which sometimes gives the wrong sequence. Therefore, a best practice is advised here to resolve the CMOS logic circuit into PULL-UP and PULL-DOWN network graphs, separately.