pathPath22002-02-03 01:56:112007-08-17 02:14:23Definitionpath length% this is the default PlanetMath preamble. as your knowledge
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% define commands hereA \emph{path} in a graph is a finite sequence of alternating vertices and edges, beginning and ending with a vertex, $v_1e_1v_2e_2v_3\dots e_{n-1}v_n$ 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., $v_1v_2v_3\dots v_n$) 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) $ABCD$ (length 3), $ABCDA$ (length 4), and $ABABABABADCBA$ (length 12). $ABD$ 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 $uev$ is a valid path but $veu$ is not.
Consider this digraph:
$$\xymatrix{
G \ar[r] \ar[d] & H \ar[d] \ar[l] \\
J & I \ar[l] }$$
$GHIJ$, $GJ$, and $GHGHGH$ are all valid paths. $GHJ$ is not a valid path because $H$ and $J$ are not connected. $GJI$ is not a valid path because the edge connecting $I$ to $J$ has the opposite orientation.