Suppose $G=(V,E)$ is a connected graph and $T=(V\_T, E\_T)$ is a subgraph of $G$ that is a tree.


If we further suppose that any pair of vertices $v,w \in V\_T$ that are not joined by a single edge in $E\_T$ are also not joined by an edge in $E$, is there a standard name for this condition? Informally, what I mean is that there are no single-edge shortcuts in the ambient graph that connect vertices in the tree that aren’t already joined by a single edge in the tree itself.



Terminology for tree subgraphs where non-neighbouring vertices are not connected by single ambient edges
The subgraph $T$ is an [induced subgraph](https://en.wikipedia.org/wiki/Induced_subgraph) of $G$.


Some authors use the term [induced tree](https://people.maths.ox.ac.uk/scott/Papers/inducedtrees.pdf) meaning an induced subgraph which is a tree.


