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Path (graph theory) 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). A directed path (sometimes called dipath[1]) in a directed graph is a finite or infinite sequence of edges which ...
Shortest path problem. Shortest path (A, C, E, D, F) between vertices A and F in the weighted directed graph. 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. [1]
A directed graph is weakly connected (or just connected [9]) if the undirected underlying graph obtained by replacing all directed edges of the graph with undirected edges is a connected graph. A directed graph is strongly connected or strong if it contains a directed path from x to y (and from y to x ) for every pair of vertices ( x , y ) .
Finally, it is clearly NP-hard on all graph classes on which the Hamiltonian path problem is NP-hard, such as on split graphs, circle graphs, and planar graphs. A simple model of a directed acyclic graph is the Price model, developed by Derek J. de Solla Price to represent citation networks. This is simple enough to allow for analytic results ...
In the mathematical field of graph theory, a Hamiltonian path (or traceable path) is a path in an undirected or directed graph that visits each edge exactly once. A Hamiltonian cycle (or Hamiltonian circuit) is a cycle that visits each vertex exactly once. A Hamiltonian path that starts and ends at adjacent vertices can be completed by adding ...
Dijkstra's algorithm (/ ˈdaɪkstrəz / DYKE-strəz) is an algorithm for finding the shortest paths between nodes in a weighted graph, which may represent, for example, road networks. It was conceived by computer scientist Edsger W. Dijkstra in 1956 and published three years later. [4][5][6] Dijkstra's algorithm finds the shortest path from a ...
The following proof for undirected graphs works without change for directed graphs or multi-graphs, provided we take path to mean directed path. For sets of vertices A,B ⊂ G (not necessarily disjoint), an AB-path is a path in G with a starting vertex in A, a final vertex in B, and no internal vertices neither in A nor in B.
The Hamiltonian path problem is a topic discussed in the fields of complexity theory and graph theory. It decides if a directed or undirected graph, G, contains a Hamiltonian path, a path that visits every vertex in the graph exactly once. The problem may specify the start and end of the path, in which case the starting vertex s and ending ...