Sewer system optimization includes two subproblems: layout optimization and hydraulic design optimization, which can be solved independently or solved simultaneously.
No matter which method is chosen for the solution of the optimization problem, a feasible layout that satisfies the restrictions of the sewer system must be obtained in any step of the solution.
Nonplanar graphs cannot be drawn on a plane or on the surface of a sphere without edges intersecting each other between the vertices.
The use of diagrams of dots and lines to represent graphs actually grew out of 19th-century chemistry, where lettered vertices denoted individual atoms and connecting lines denoted chemical bonds (with degree corresponding to valence), in which planarity had important chemical consequences.
Our editors will review what you’ve submitted and determine whether to revise the article.
Join Britannica's Publishing Partner Program and our community of experts to gain a global audience for your work! The Königsberg bridge problem was an old puzzle concerning the possibility of finding a path over every one of seven bridges that span a forked river flowing past an island—but without crossing any bridge twice. His proof involved only references to the physical arrangement of the bridges, but essentially he proved the first theorem in graph theory.degree, which is defined as the number of edges that enter or exit from it.The knight’s tour ( number game: Chessboard problems) is another example of a recreational problem involving a Hamiltonian circuit.Hamiltonian graphs have been more challenging to characterize than Eulerian graphs, since the necessary and sufficient conditions for the existence of a Hamiltonian circuit in a connected graph are still unknown.This work confirmed that a formula of the English mathematician Percy Heawood from 1890 correctly gives these colouring numbers for all surfaces except the one-sided surface known as the Klein bottle, for which the correct colouring number had been determined in 1934.traveling salesman problem (the shortest path that begins and ends at the same vertex and visits each edge exactly once), which continues to attract the attention of many researchers because of its applications in routing data, products, and people.Work on such problems is related to the field of linear programming, which was founded in the mid-20th century by the American mathematician George Dantzig.These are graphs that can be drawn as dot-and-line diagrams on a plane (or, equivalently, on a sphere) without any edges crossing except at the vertices where they meet.Complete graphs with four or fewer vertices are planar, but complete graphs with five vertices () or more are not. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.A graph theory-based methodology is proposed for the sewer system optimization problem in this study.If there is a path linking any two vertices in a graph, that graph is said to be connected.A path that begins and ends at the same vertex without traversing any edge more than once is called a A graph is a collection of vertices, or nodes, and edges between some or all of the vertices.