I just started learning graph theory and I have the following definitions:
Def:
A directed graph or digraph G consists of disjoint finite sets V = V(G)of nodes and E = E(G) of arcs, and functions associating to each $e\in E$ a tail $t(e) \in V$ and a head $h(e) \in V$.
Def:
When we write $e = vw$ for an arc of a digraph G, we mean that $v = t(e)$, $w = h(e)$. An arc $e_i$ of a path $P: v_0, e_1, v_1, ..., e_k, v_k$ is forward if $t(e_i) = v_{i-1}$ and $h(e_i) = v_i$ and is reverse otherwise.
Question: I don't fully understand the second definition. If an arc in a path is forward if $t(e_i) = v_{i-1}$ and $h(e_i) = v_i$, isn't an arc in a path always forward? Since the path moves from $v_0$ to $v_1$, from $v_1$ to $v_2$, etc?
Thanks in advance!
In a directed path, that ought to be true. It seems like you might be considering an undirected path in the directed graph: that is, a sequence of vertices that, if we suppress all edge orientations, becomes a path in the undirected graph we get.
This is sometimes useful, and in this case, the undirected path can indeed have forward and reverse arcs along it.