<P> In graph theory, a path in a graph is a finite or infinite sequence of edges which connect a sequence of vertices which, by most definitions, are all distinct from one another . In a directed graph, a directed path (sometimes called dipath) is again a sequence of edges (or arcs) which connect a sequence of vertices, but with the added restriction that the edges all be directed in the same direction . </P> <P> Paths are fundamental concepts of graph theory, described in the introductory sections of most graph theory texts . See e.g. Bondy and Murty (1976), Gibbons (1985), or Diestel (2005). Korte et al. (1990) cover more advanced algorithmic topics concerning paths in graphs . </P> <P> A path is a trail in which all vertices (except possibly the first and last) are distinct . A trail is a walk in which all edges are distinct . A walk of length k (\ displaystyle k) in a graph is an alternating sequence of vertices and edges, v 0, e 0, v 1, e 1, v 2,..., v k − 1, e k − 1, v k (\ displaystyle v_ (0), e_ (0), v_ (1), e_ (1), v_ (2), \ ldots, v_ (k - 1), e_ (k - 1), v_ (k)), which begins and ends with vertices . If the graph is undirected, then the endpoints of e i (\ displaystyle e_ (i)) are v i (\ displaystyle v_ (i)) and v i + 1 (\ displaystyle v_ (i + 1)). If the graph is directed, then e i (\ displaystyle e_ (i)) is an arc from v i (\ displaystyle v_ (i)) to v i + 1 (\ displaystyle v_ (i + 1)). An infinite path is an alternating sequence of the same type described here, but with no first or last vertex, and a semi-infinite path (also ray) has a first vertex, v 0 (\ displaystyle v_ (0)), but no last vertex . Most authors require that all of the edges and vertices be distinct from one another . However, some authors do not make this requirement, and instead use the term simple path to refer to a path which contains no repeated vertices . </P> <P> 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 . </P>

Definition of walk in graph theory with examples
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