Notation for path in graph Let $G=(V,E)$ be a graph with vertices $V$ and edges $E$.
Is there standard notation for stating that $\pi$ is a path in $G$?
I though about these options:


*

*$\pi \in G$

*$\pi \sim G$

*$\pi \in \text{Paths}(G)$
However, I do not remember seeing this notation used anywhere. Is there more standard notation for this?
 A: There is no widely used notation. In general, concepts such as "the set of all paths", "the concatenation of paths", "the reverse of a path", "the shortest path from $s$ to $t$", and so forth are typically expressed in words rather than in mathematical notation.
You should not use notation such as $\pi \in G$ because "$\in$" (is an element of) already has a meaning in mathematical notation, and being a path in $G$ is not that meaning. A graph is not a set of paths. (Usually, a graph is an ordered pair $(V,E)$, which means that viewing it as any kind of set at all is foundation-dependent.) Similarly, $\pi \sim G$ is not great; there are a few things that $\sim$ suggests, mostly equivalence relations, and "is a path of" is not like any of them.
On the other hand, defining a set such as $\text{Paths}(G)$ (or $P(G)$, $\mathcal P(G)$, $\mathsf{PATHS}(G)$, etc.) is not unusual: it's not standard enough that you should use it without a definition, but it's common enough that nobody will be surprised.
It is much more typical in graph theory to explain what you mean in words rather than talking about the set of all paths. (As a graph theorist, I have the arrogant belief that this is because we like proofs that have meaning, as opposed to proofs that are just pushing symbols around like you get in some other areas of math.) I would avoid defining 
$\text{Paths}(G)$ unless you actually need to do something for every single path. 
For example, I can imagine a linear program with a variable $x_\pi$ for every path $\pi$; that's one case where bringing up $\text{Paths}(G)$ might make the explanation more clear. On the other hand, "the set of all path lengths", the example you give, is not a case where that seems necessary. Neither is $|\pi|$ obviously the length of a path; a path of length $k$ has $k$ edges and $k+1$ vertices and so I can imagine $|\pi|$ being either $k$ or $k+1$ or even (in a technical way that you'd rarely actually want) $2k+1$.
