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(Translated into the terminology of modern graph theory, Euler’s theorem about the Königsberg bridge problem could be restated as follows: If there is a path along edges of a multigraph that traverses each edge once and only once, then there exist at most two vertices of odd degree; furthermore, if the path begins and ends at the same vertex, then no vertices will have odd degree.)path, which is any route along the edges of a graph.A path may follow a single edge directly between two vertices, or it may follow multiple edges through multiple vertices.Our editors will review what you’ve submitted and determine whether to revise the article.
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A graph theory-based methodology is proposed for the sewer system optimization problem in this study.
Thus, a loop contributes 2 to the degree of its vertex.
For instance, the vertices of the simple graph shown in the diagram all have a degree of 2, whereas the vertices of the complete graph shown are all of degree 3.
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.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.Having considered a surface divided into polygons by an embedded graph, mathematicians began to study ways of constructing surfaces, and later more general spaces, by pasting polygons together.This was the beginning of the field of combinatorial topology, which later, through the work of the French mathematician Henri Poincaré and others, grew into what is known as algebraic topology.planar graphs.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.When there exists a path that traverses each edge exactly once such that the path begins and ends at the same vertex, the path is known as an Eulerian circuit, and the graph is known as an Eulerian graph.Hamiltonian circuit, along the edges of a dodecahedron (a Platonic solid consisting of 12 pentagonal faces) that begins and ends at the same corner while passing through each corner exactly once.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.