edge-induced subgraph

let $\text{[math]}$ be an undirected graph, and let $\text{[math]}$ .
an edge-induced subgraph is a graph that contains the edges $\text{[math]}$ and all the vertices at the endpoints of these edges.

let $\text{[math]}$ be a connected undirected graph, assume that bfs has been applied to $\text{[math]}$ starting at the vertex $\text{[math]}$ . let $\text{[math]}$ be the edge-induced subgraph with the edges
$\text{[math]}$ the resulting tree $\text{[math]}$ is a distance tree relative to $\text{[math]}$ in $\text{[math]}$ .

with the help of this bfs lemma:
each vertex in the graph is found once, so $\text{[math]}$ has $\text{[math]}$ edges and $\text{[math]}$ vertices, for each vertex $\text{[math]}$ , in $\text{[math]}$ there would be a path from $\text{[math]}$ to $\text{[math]}$ , so $\text{[math]}$ is a tree by definition. for all $\text{[math]}$ , the only path from $\text{[math]}$ to $\text{[math]}$ in $\text{[math]}$ is a minimal path in $\text{[math]}$ , i.e. $\text{[math]}$ .