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# path

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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.

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.

## Mathematics Subject Classification

05C38*no label found*

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