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.



Shortcutting quasigeodesics
As suggested, I am turning my comments into an answer. The answer to both questions is negative for any $\lambda > 1$, and positive for $\lambda = 1$.


For $\lambda = 1+\epsilon$ note that in $\mathbb Z^2$, the concatenation of the (unique) geodesics from $(0,0)$ to $(0,k)$, from $(0,k)$ to $(n,k)$ and from $(n,k)$ to $(n,0)$ is $(1+\epsilon,0)$-quasigeodesic given $\frac{2k}{n} < \epsilon$.


If we replace the segment between $(0,0)$ and $(n-1,k)$ by the geodesic from $(0,0)$ via $(n-1,0)$ to $(n-1,k)$ then the points $(0,n-1)$ and $(0,n)$ have distance $1$ in $\mathbb Z^2$, but distance $2k+1$ on the path. So this path can't be $(\lambda',c')$-quasigeodesic unless $\lambda'+c' \geq 2k+1$.


For $\lambda = 1$, note that a path $p$ is $(1,c)$-quasigeodesic if and only if $\ell(p) \leq d(p\_-,p\_+)+c$; in other words, it suffices to check the condition for the whole path rather than for all sub-paths. This property is clearly preserved when replacing any subpath by a shorter or equally long one.


