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  2. Eulerian path - Wikipedia

    en.wikipedia.org/wiki/Eulerian_path

    A connected graph has an Euler cycle if and only if every vertex has an even number of incident edges. The term Eulerian graph has two common meanings in graph theory. One meaning is a graph with an Eulerian circuit, and the other is a graph with every vertex of even degree. These definitions coincide for connected graphs. [2]

  3. BEST theorem - Wikipedia

    en.wikipedia.org/wiki/BEST_theorem

    In 1736, Euler showed that G has an Eulerian circuit if and only if G is connected and the indegree is equal to outdegree at every vertex. In this case G is called Eulerian. We denote the indegree of a vertex v by deg(v). The BEST theorem states that the number ec(G) of Eulerian circuits in a connected Eulerian graph G is given by the formula

  4. Seven Bridges of Königsberg - Wikipedia

    en.wikipedia.org/wiki/Seven_Bridges_of_Königsberg

    Map of Königsberg in Euler's time showing the actual layout of the seven bridges, highlighting the river Pregel and the bridges. The Seven Bridges of Königsberg is a historically notable problem in mathematics. Its negative resolution by Leonhard Euler, in 1736, [1] laid the foundations of graph theory and prefigured the idea of topology. [2]

  5. List of graph theory topics - Wikipedia

    en.wikipedia.org/wiki/List_of_graph_theory_topics

    Path (graph theory) Seven Bridges of Königsberg. Eulerian path; Three-cottage problem; Shortest path problem. Dijkstra's algorithm. Open Shortest Path First; Flooding algorithm; Route inspection problem; Hamiltonian path. Hamiltonian path problem; Knight's tour; Traveling salesman problem. Nearest neighbour algorithm; Bottleneck traveling ...

  6. Cycle basis - Wikipedia

    en.wikipedia.org/wiki/Cycle_basis

    A spanning subgraph of a given graph G has the same set of vertices as G itself but, possibly, fewer edges. A graph G, or one of its subgraphs, is said to be Eulerian if each of its vertices has even degree (its number of incident edges). Every simple cycle in a graph is an Eulerian subgraph, but there may be others.

  7. Euler tour technique - Wikipedia

    en.wikipedia.org/wiki/Euler_tour_technique

    The Euler tour technique (ETT), named after Leonhard Euler, is a method in graph theory for representing trees. The tree is viewed as a directed graph that contains two directed edges for each edge in the tree. The tree can then be represented as a Eulerian circuit of the directed graph, known as the Euler tour representation (ETR) of the tree

  8. Graph theory - Wikipedia

    en.wikipedia.org/wiki/Graph_theory

    In mathematics and computer science, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of vertices (also called nodes or points ) which are connected by edges (also called arcs , links or lines ).

  9. Chinese postman problem - Wikipedia

    en.wikipedia.org/wiki/Chinese_postman_problem

    When the graph has an Eulerian circuit (a closed walk that covers every edge once), that circuit is an optimal solution. Otherwise, the optimization problem is to find the smallest number of graph edges to duplicate (or the subset of edges with the minimum possible total weight) so that the resulting multigraph does have an Eulerian circuit. [1]