  For a graph $G=(V,E)$ and positive integer $p$, the exact distance-$p$ graph
$G^{[\natural p]}$ is the graph with vertex set $V$ and with an edge between
vertices $x$ and $y$ if and only if $x$ and $y$ have distance $p$. Recently,
there has been an effort to obtain bounds on the chromatic number
$\chi(G^{[\natural p]})$ of exact distance-$p$ graphs for $G$ from certain
classes of graphs. In particular, if a graph $G$ has tree-width $t$, it has
been shown that $\chi(G^{[\natural p]}) \in \mathcal{O}(p^{t-1})$ for odd $p$,
and $\chi(G^{[\natural p]}) \in \mathcal{O}(p^{t}\Delta(G))$ for even $p$. We
show that if $G$ is chordal and has tree-width $t$, then $\chi(G^{[\natural
p]}) \in \mathcal{O}(p\, t^2)$ for odd $p$, and $\chi(G^{[\natural p]}) \in
\mathcal{O}(p\, t^2 \Delta(G))$ for even $p$.
  If we could show that for every graph $H$ of tree-width $t$ there is a
chordal graph $G$ of tree-width $t$ which contains $H$ as an isometric subgraph
(i.e., a distance preserving subgraph), then our results would extend to all
graphs of tree-width $t$. While we cannot do this, we show that for every graph
$H$ of genus $g$ there is a graph $G$ which is a triangulation of genus $g$ and
contains $H$ as an isometric subgraph.
