The vertex set of a connected simple graph with its path metric is a $(1,1)$-quasi-geodesic space

Given: The hypotheses of the Statement.

A $(c,b)$-quasi-geodesic is a $(c,b)$-quasi-isometric embedding of a closed real interval, and a space is $(c,b)$-quasi-geodesic when every two of its points are joined by one (Quasi-geodesics and quasi-geodesic metric spaces).

The path metric of a connected simple graph assigns to two vertices the least length of a path joining them (The path metric of a connected simple graph).

The path metric of a connected simple graph is a metric on its vertex set (The path metric of a connected simple graph is a metric on its vertex set).

A geodesic of length $L$ in a metric space is an isometric embedding of the interval $[0,L]$, and the space is geodesic when every two points are the endpoints of one (Geodesics and geodesic metric spaces).

A map is $(L,C)$-coarse Lipschitz when $d(f(x),f(x^{\mathrm{\prime }}))\le Ld(x,x^{\mathrm{\prime }})+C$, and an $(L,C)$-quasi-isometric embedding when in addition $L^{-1}d(x,x^{\mathrm{\prime }})-C\le d(f(x),f(x^{\mathrm{\prime }}))$ (Coarse Lipschitz maps and quasi-isometric embeddings).

It is written $\lfloor x\rfloor$ and called the integer part, or floor, of $x$. (Integer part: for every real $x$ there is exactly one integer $m$ with $m\le x<m+1$).