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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 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).
Conversely, if H has an induced path or cycle of length k, any maximal set of nonadjacent vertices in G from this path or cycle forms an independent set in G of size at least k/3. Thus, the size of the maximum independent set in G is within a constant factor of the size of the longest induced path and the longest induced cycle in H.
A Hamiltonian path or traceable path is a path that visits each vertex of the graph exactly once. A graph that contains a Hamiltonian path is called a traceable graph.A graph is Hamiltonian-connected if for every pair of vertices there is a Hamiltonian path between the two vertices.
The general graph Steiner tree problem can be approximated by computing the minimum spanning tree of the subgraph of the metric closure of the graph induced by the terminal vertices, as first published in 1981 by Kou et al. [18] The metric closure of a graph G is the complete graph in which each edge is weighted by the shortest path distance ...
A planar graph and its minimum spanning tree. Each edge is labeled with its weight, which here is roughly proportional to its length. A minimum spanning tree (MST) or minimum weight spanning tree is a subset of the edges of a connected, edge-weighted undirected graph that connects all the vertices together, without any cycles and with the minimum possible total edge weight. [1]
In it, any two points can reach each other along a path through the line segments. It can be found as the minimum spanning tree of a complete graph with the points as vertices and the Euclidean distances between points as edge weights. The edges of the minimum spanning tree meet at angles of at least 60°, at most six to a vertex.
If this check passes, next, the algorithm will ensure that the first vertex in c is equal to s and the last vertex is equal to t. Lastly, to verify that c is a valid path, the algorithm must check that every edge between vertices in c is indeed an edge in G. If any of these checks fail, the algorithm will reject. Otherwise, it will accept. [22 ...
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