Computing minimum rainbow and strong rainbow colorings of block graphs

Abstract

<p>A path in an edge-colored graph G is rainbow if no two edges of it are colored the same. The graph G is rainbowconnected 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 ofG, 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 n<sup>1-2-ϵ</sup> 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.</p>