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 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
src(G). We prove that for every k≥3, deciding whether
src(G)≤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 n1/2−ϵ for any ϵ>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.Comment: 13 pages, 3 figure