Dependence on the power parameter

Determine whether the dependence on the integer power parameter $r$ is necessary in the upper bound for the strong isometric path complexity of graph powers given in Theorem 3.1(a).

Background

Theorem 3.1(a) bounds the isometric and strong isometric path complexities of the $r$th power $G^r$ by a quantity proportional to $k(4r^2-2r)$ when the corresponding complexity of $G$ is at most $k$. The authors observe that for powers of chordal graphs with even $r$, no dependence on $r$ is needed, motivating the question of whether the general dependence is an artifact of the proof or genuinely necessary.