$r+1$, $s_1+1$, and $s_2+1$, where the internal vertices of the paths are $2{{\ensuremath{^\perp}}}$-vertices. The paths of lengths $t_1+1$ and $t_2+1$ have $v$ as an endpoint, the path of length $r+1$ has $u$ and $v$ as endpoints and the paths of lengths $s_1+1$ and $s_2+1$ have $u$ as an endpoint. We call such configuration an *$H$-type configuration about $v$ and $u$*, see Figure \[fig:H\].

A configuration $\Phi(t,a_1,a_2,s)$ is formed by two 3-vertices $u$ and $v$ and 4 internally disjoint paths of lengths $t+1$, $a_1+1$, $a_2+1$, and $s+1$, where the internal vertices of the paths are $2{{\ensuremath{^\perp}}}$-vertices. The path of length $t+1$ has $v$ as an endpoint, the paths of lengths $a_1+1$ and $a_2+1$ have $u$ and $v$ as endpoints and the path of length $s+1$ has $u$ as an endpoint. We call such configuration a *$\Phi$-type configuration about $v$ and $u$*, see Figure \[fig:Phi\].

The reducibility of these configurations was verified using computer[^3], and in addition the 8-caterpillar is addressed in Lemma \[lma:caterpillar\]. Given the definition of a $2{{\ensuremath{^\perp}}}$-vertex, the vertices of degree two in these configurations may, or may not, be adjacent to some 1-vertices in $G$. We demonstrate the reducibility of the instances of these configurations wherein each vertex 