I need help understanding this Lemma and proof.
Graph theory - identifying vertex-lema.png


$ Lemma \text{ } \\
Let \text{ }G \text{ be the graph obtained from a graph } G_{1} \text{by identifying an arbitrary vertex of } G_{1} \\ \text{ and one pendent vertex of the path } P_{2} \text{.} \\ \text{Then the determinant of the distance matrix of the graph } G \text{ is fixed, regardless the} \\ \text{choice of the vertex of }
G\text{.}$


Does $P_{2}$ mean chordless path on $2$ vertices?
Does identifying vertices mean contracting them in one vertex which has edges to all vertices that these two had?
How is it possible that then $G_{1}$ is subgraph of $G$ when $G$ was obtained from $G_{1}$ by identifying vertices?

I'm probably getting something wrong here...

It's from here https://arxiv.org/pdf/1308.2281.pdf