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In graph theory, Turán's theorem bounds the number of edges that can be included in an undirected graph that does not have a complete subgraph of a given size. It is one of the central results of extremal graph theory, an area studying the largest or smallest graphs with given properties, and is a special case of the forbidden subgraph problem on the maximum number of edges in a graph that ...
Shortest path problem. In graph theory, the shortest path problem is the problem of finding a path between two vertices (or nodes) in a graph such that the sum of the weights of its constituent edges is minimized. The problem of finding the shortest path between two intersections on a road map may be modeled as a special case of the shortest ...
In graph theory, a branch of mathematics and computer science, Guan's route problem, the Chinese postman problem, postman tour or route inspection problem is to find a shortest closed path or circuit that visits every edge of an (connected) undirected graph at least once. When the graph has an Eulerian circuit (a closed walk that covers every ...
Turn-by-turn systems typically use an electronic voice to inform the user whether to turn left or right, the street name, and the distance to the next turn. [3] Mathematically, turn by turn navigation is based on the shortest path problem within graph theory , which examines how to identify the path that best meets some criteria (shortest ...
A graph that is an optimal solution to such an optimization problem is called an extremal graph, and extremal graphs are important objects of study in extremal graph theory. Extremal graph theory is closely related to fields such as Ramsey theory, spectral graph theory, computational complexity theory, and additive combinatorics, and frequently ...
Forbidden subgraph problem. In extremal graph theory, the forbidden subgraph problem is the following problem: given a graph , find the maximal number of edges an -vertex graph can have such that it does not have a subgraph isomorphic to . In this context, is called a forbidden subgraph. [1]
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.
Solution of a travelling salesperson problem: the black line shows the shortest possible loop that connects every red dot. The travelling salesman problem, also known as the travelling salesperson problem (TSP), asks the following question: "Given a list of cities and the distances between each pair of cities, what is the shortest possible route that visits each city exactly once and returns ...