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In mathematics, a path in a graph is a sequence of vertices such that from each of its vertices there is an edge to the successor vertex.
A path is called simple path if none of the vertices in the path are repeated.
Two paths are independent if they do not have any vertex in common, except the first and last one.
The length of a path is the number of edges that the path uses, counting multiple edges multiple times. In the example graph, (1, 2, 5, 1, 2, 3) is a path with length 5, and (5, 2, 1) is a simple path of length 2.
A weighted graph associates a value (weight) with every edge in the graph. The weight of a path in a weighted graph is the sum of the weights of the traversed edges. Sometimes the words cost or length are used instead of weight.
If it is possible to establish a path from any vertex to any other vertex of a graph, the graph is said to be connected. If it is always possible to establish a path from any vertex to any other vertex even after removing k-1 vertices, then the graph is said to be k-connected. Note that a graph is k-connected if and only if it contains k independent paths between any two vertices.
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