  A path in an edge-colored graph $G$ is rainbow if no two edges of it are
colored the same. The graph $G$ is rainbow-connected if there is a rainbow path
between every pair of vertices. If there is a rainbow shortest path between
every pair of vertices, the graph $G$ is strongly rainbow-connected. The
minimum number of colors needed to make $G$ rainbow-connected is known as the
rainbow connection number of $G$, and is denoted by $\text{rc}(G)$. Similarly,
the minimum number of colors needed to make $G$ strongly rainbow-connected is
known as the strong rainbow connection number of $G$, and is denoted by
$\text{src}(G)$. We prove that for every $k \geq 3$, deciding whether
$\text{src}(G) \leq k$ is NP-complete for split graphs, which form a subclass
of chordal graphs. Furthermore, there exists no polynomial-time algorithm for
approximating the strong rainbow connection number of an $n$-vertex split graph
with a factor of $n^{1/2-\epsilon}$ for any $\epsilon > 0$ unless P = NP. We
then turn our attention to block graphs, which also form a subclass of chordal
graphs. We determine the strong rainbow connection number of block graphs, and
show it can be computed in linear time. Finally, we provide a polynomial-time
characterization of bridgeless block graphs with rainbow connection number at
most 4.
