Let $G$ be a graph (i.e., an undirected graph in which we allow for loops and parallel edges). Denote by $V$ the vertex set, by $E$ the edge set, and by $\psi$ the incidence function of $G$, and let $\mathscr F(X)$ be the free monoid over a set $X$. I'll write $\ast_X$ for the operation of (word) concatenation in $\mathscr F(X)$ and omit the subscript $X$ if there is no risk of confusion.
Formally speaking, a walk (in the graph $G$) is a non-empty word $\mathfrak z = z_0 \ast \cdots \ast z_k \in \mathscr F(V \cup E)$ of odd length $k+1$ such that
- $z_{2i} \in V$ for each $i \in [\![0, k/2 ]\!]$, and
- $z_{2i+1} \in E$ and $\psi(z_{2i+1}) = \{z_{2i}, z_{2i+2}\}$ for each $i \in [\![0, k/2-1 ]\!]$.
Just in case, note that the length of $\mathfrak z$ as a $(V \cup E)$-word is not the same as the length of $\mathfrak z$ as a walk, which, by definition, is rather equal to $k/2$.
Now, the concatenation $\mathfrak u \ast \mathfrak v$ of a walk $\mathfrak u = u_0 \ast \cdots \ast u_h$ of length $h$ with a walk $\mathfrak v = v_0 \ast \cdots \ast v_k$ of length $k$ is not a walk (regardless of any restrictions on $\mathfrak u$ and $\mathfrak v$). However, if $(\mathfrak u, \mathfrak v)$ is a pair of composable walks in the sense that the origin $v_0$ of $\mathfrak v$ and the terminus $u_h$ of $\mathfrak v$ coincide, then the concatenation of $\mathfrak u$ with the subword of $\mathfrak v$ obtained from pruning $v_0$ off of $\mathfrak v$ is a walk (with origin $u_0$ and terminus $v_k$), herein denoted by $\mathfrak u \circ_G \mathfrak v$ and called the $G$-composition of $\mathfrak u$ with $\mathfrak v$.
Question. Is there a standard name for the partial (binary) operation that maps a pair of composable walks to their $G$-composition? Is there a standard notation for this composition? And is there any textbook on graph theory where things are worked out more or less in these terms?
"Composition" sounds like a natural name to me, since the walks of the graph $G$ can be viewed as the morphisms of a category whose objects are the vertices of $G$ and where the composition law is precisely the partial operation that maps a pair of composable walks to their $G$-composition. But I'd like to hear from the graph theorists.
