edge contraction

for a graph $\text{[math]}$ and an edge $\text{[math]}$ , the contraction of $\text{[math]}$ in $\text{[math]}$ is an operation that produces the graph $\text{[math]}$ , where

$\text{[math]}$ , and
the part of $\text{[math]}$ containing $\text{[math]}$ and the part of $\text{[math]}$ containing $\text{[math]}$ are merged (and $\text{[math]}$ is removed) to form a part $\text{[math]}$ .

the graph obtained from $\text{[math]}$ by contracting $\text{[math]}$ is denoted $\text{[math]}$ .

like deletions, the order of contractions of edges does not affect the result. for a set $\text{[math]}$ of edges, the graph obtained by contracting the edges of $\text{[math]}$ is denoted $\text{[math]}$ .
[cite:;taken from @graph_klein_2024 chapter 2.2.5 contraction of edges]