Is there such a thing like a one-side (meaning just one of the sets $V_1$ or $V_2$) component of a bipartite graph $B = (V_1, V_2, E)$?
For example, for such a bipartite graph $B = (V_1, V_2, E)$, I am interested in the component(s) only for nodes from $V_1$, the vertices from $V_2$ are only relevant for the connection. In other words, only the neighbors of neighbors of nodes in $V_1$ are interesting for me.
Best Answer
I have seen this notion under the name "2-linked" components, typically when discussing graph container arguments like David Galvin's "Independent sets in the discrete hypercube".
One definition of the $k^{\text{th}}$ power $G^k$ of a graph $G$ is the graph on $V(G)$ with an edge between two vertices if their distance in $G$ is at most $k$. See On the Pósa-Seymour conjecture by Komlós, Sárközy, and Szemerédi for an example of this usage with Hamiltonian cycles. We then say that a set $S$ of vertices is $k$-linked if $G^k[S]$, the subgraph of $G^k$ induced by $S$, is connected.
Since you are explicitly working with bipartite graphs, you might find it easier to just write things in terms of neighborhoods. Let $N(v)$ be the neighborhood of $v$ in an underlying bipartite graph $G$, and let $N(S)$ be the union of $N(v)$ for all $v \in S$. Then $N(N(v))$ is the set of all vertices at distance $0$ or $2$ from $v$. You can define a component $A$ as a minimal set of vertices for which $N(N(A)) = A$.