Connected graph is a quasi-geodesic space

Question

I am trying to understand why if $X=(V,E)$ is a connected graph, then the associated metric on $V$ makes that $V$ can be see as $(1,1)-$quasi-geodesic space, I understand the notion of what a quasi-geodesic space means, but I dont get why this space need the $(1,1)$. Thanks

Answer 1

If $x,y\in V$, let $x=x_0,x_1,\ldots,x_n=y$ be a path connecting $x$ and $y$ of minimal length. Define $\gamma:[0,n]\to V$ by $\gamma(t)=x_{[t]}$, where $[t]$ denotes the integer part of $t$. This is a $(1,1)$-quasi-geodesic. For if $t_0,t_1\in[0,n]$, with $t_0\leq t_1$ we have $d(\gamma(t_0),\gamma(t_1))=[t_1]-[t_0]$, and $$(t_1-t_0)-1\leq[t_1]-[t_0]\leq(t_1-t_0)+1.$$