path

\\xyoption

all

A path in a graph is a finite sequence of alternating vertices and edges, beginning and ending with a vertex, $v_{1}e_{1}v_{2}e_{2}v_{3}\\dots e_{n-1}v_{n}$ such that every consecutive pair of vertices $v_{x}$ and $v_{x+1}$ are adjacent and $e_{x}$ is incident with $v_{x}$ and with $v_{x+1}$ . Typically, the edges may be omitted when writing a path (e.g., $v_{1}v_{2}v_{3}\\dots v_{n}$ ) since only one edge of a graph may connect two adjacent vertices. In a multigraph, however, the choice of edge may be significant.

The length of a path is the number of edges in it.

Consider the following graph:

\\xymatrix{A\\ar@{-}[r]&B\\ar@{-}[d]\\\\D\\ar@{-}[u]&C\\ar@{-}[l]}

Paths include (but are certainly not limited to) $A B C D$ (length 3), $A B C D A$ (length 4), and $A B A B A B A B A D C B A$ (length 12). $A B D$ is not a path since $B$ is not adjacent to $D$ .

In a digraph, each consecutive pair of vertices must be connected by an edge with the proper orientation; if $e=(u,v)$ is an edge, but $(v,u)$ is not, then $u e v$ is a valid path but $v e u$ is not.

Consider this digraph:

\\xymatrix{G\\ar[r]\\ar[d]&H\\ar[d]\\ar[l]\\\\J&I\\ar[l]}

$G H I J$ , $G J$ , and $G H G H G H$ are all valid paths. $G H J$ is not a valid path because $H$ and $J$ are not connected. $G J I$ is not a valid path because the edge connecting $I$ to $J$ has the opposite orientation.