Let $\Gamma$ be a connected graph, let $\lambda \ge 1$ and $c \ge 0$ be some constants. Recall that a combinatorial path $p$ in $\Gamma$ is said to be $(\lambda,c)$-quasigeodesic if for every combinatorial subpath $q$ of $p$ one has $$\ell(q) \le \lambda d(q_-,q_+)+c,$$ where $\ell(q)$ is the length of $q$, $q_-$ and $ q_+$ are the endpoints of $q$, and $d(\cdot,\cdot)$ is the standard metric on $\Gamma$.
Question 1: suppose that a path $s$ in $\Gamma$ has been obtained from a $(\lambda,c)$-quasigeodesic path $p$ by replacing some (combinatorial) subpaths of $p$ with geodesics. Is it true that this new path $s$ is again $(\lambda,c)$-quasigeodesic?
Intuitively, the answer should be "Yes", because replacing subpaths with geodesics ("shortcutting") should only improve the quasigeodesicity constants. However, I do not see how to prove this.
If the answer to Question 1 is negative, the natural next question is the following:
Question 2: suppose that a path $s$ in $\Gamma$ has been obtained from a $(\lambda,c)$-quasigeodesic path $p$ by replacing some subpaths of $p$ with geodesics. Is it true that this new path $s$ is $(\lambda',c')$-quasigeodesic, where the constants $\lambda' \ge 1$, $c' \ge 0$ depend only on $\Gamma$, $\lambda$ and $c$?
It's not hard to show that the answer to Question 2 is positive when the graph $\Gamma$ is $\delta$-hyperbolic, but I do not have a proof or a counter-example for more general graphs.