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In graph theory and theoretical computer science, the longest path problem is the problem of finding a simple path of maximum length in a given graph.A path is called simple if it does not have any repeated vertices; the length of a path may either be measured by its number of edges, or (in weighted graphs) by the sum of the weights of its edges.
A bipartite graph may be oriented from one side of the bipartition to the other. The longest path in this orientation has length one, with only two vertices. Conversely, if a graph is oriented without any three-vertex paths, then every vertex must either be a source (with no incoming edges) or a sink (with no outgoing edges) and the partition of the vertices into sources and sinks shows that ...
Grundy number of a directed graph. [3]: GT56 Hamiltonian completion [3]: GT34 Hamiltonian path problem, directed and undirected. [2] [3]: GT37, GT38, GT39 Induced subgraph isomorphism problem; Graph intersection number [3]: GT59 Longest path problem [3]: ND29 Maximum bipartite subgraph or (especially with weighted edges) maximum cut.
A three-dimensional hypercube graph showing a Hamiltonian path in red, and a longest induced path in bold black. In graph theory, a path in a graph is a finite or infinite sequence of edges which joins a sequence of vertices which, by most definitions, are all distinct (and since the vertices are distinct, so are the edges).
In the symmetric TSP, the distance between two cities is the same in each opposite direction, forming an undirected graph. This symmetry halves the number of possible solutions. In the asymmetric TSP, paths may not exist in both directions or the distances might be different, forming a directed graph. Traffic congestion, one-way streets, and ...
An induced path of length four in a cube.Finding the longest induced path in a hypercube is known as the snake-in-the-box problem.. In the mathematical area of graph theory, an induced path in an undirected graph G is a path that is an induced subgraph of G.
The graph shown has one maximum clique, the triangle {1,2,5}, and four more maximal cliques, the pairs {2,3}, {3,4}, {4,5}, and {4,6}. An undirected graph is formed by a finite set of vertices and a set of unordered pairs of vertices, which are called edges.
To decide if a graph has a Hamiltonian path, one would have to check each possible path in the input graph G. There are n! different sequences of vertices that might be Hamiltonian paths in a given n-vertex graph (and are, in a complete graph), so a brute force search algorithm that tests all possible sequences would be very slow.