Let $n$ be a positive integer and $K$ be the set of all the $2$-elements subsets of $\{1,2,...,n\}$,then $|K|= \binom{n}{2}$. Define $$S=\{P\subseteq K:\bigcup_{I\in P}I=\{1,2,...,n\}\}.$$

For any $P\in S$,define a simple graph $G_P$:

$(1)$$G_P$ has $n+|P|$ vertices and $V(G_P)=\{v_i:i=1,2,...,n\}\bigcup\{w_{\{j,k\}}:\{j,k\}\in P\}$;

$(2)$$E(G_P)=\{v_iv_{i+1}:i=1,2,...,n-1\}\bigcup\{w_{\{j,k\}}v_j,w_{\{j,k\}}v_k:\{j,k\}\in P\}$.

Obviously there has a path between $v_1$ and $v_n$ in $G_P$ whose length is $n-1$.I guess for any $P\in S$,there must exist another path between $v_1$ and $v_n$ in $G_P$ whose length is larger than $n-1$,is it true？