Recent developments have added weighted connections to create a weighted graph so that causal analysis or Causality is possible for the represented value chains.
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The reduction takes as input an instance of the Steiner tree problem : a weighted graph, with a subset of its vertices selected as terminals.
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The stretch factor is important in the theory of geometric spanners, weighted graphs that approximate the Euclidean distances between a set of points in the Euclidean plane.
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For example, a minimum spanning tree of a weighted graph may be obtained using Kruskal's algorithm, which is a greedy algorithm for the cycle matroid.
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Best-first search algorithms, like the A * search algorithm, find the shortest path between two nodes of a weighted graph, trying out the most promising routes first.
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:: Updating the previous comment . . . If you can represent all options as a weighted graph, you can use Dijkstra's algorithm to quickly find an optimal solution.
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Various generalizations of line graphs have also been studied, including the line graphs of line graphs, line graphs of multigraphs, line graphs of hypergraphs, and line graphs of weighted graphs.
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(5 ) For any fixed rational number t > 1, it is NP-complete to determine whether a weighted graph contains a tree t-spanner, even if all edge weights are positive integers.
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If " G " is a bipartite multigraph or weighted graph then the elements " b " " i ", " j " are taken to be the number of edges between the vertices or the weight of the edge, respectively.
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Common formulations of the clique problem include finding a maximum clique ( a clique with the largest possible number of vertices ), finding a maximum weight clique in a weighted graph, listing all maximal cliques ( cliques that cannot be enlarged ), and solving the decision problem of testing whether a graph contains a clique larger than a given size.
