Given a vertex-colored graph, we say a path is a rainbow vertex path if all its internal vertices have distinct colors. The graph is rainbow vertex-connected if there is a rainbow vertex path between every pair of its vertices. In the Rainbow Vertex Coloring (RVC) problem we want to decide whether the vertices of a given graph can be colored with at most k colors so that the graph becomes rainbow vertex-connected. This problem is known to be NP-complete even in very restricted scenarios, and very few efficient algorithms are known for it. In this work, we give polynomial-time algorithms for RVC on permutation graphs, powers of trees and split strongly chordal graphs. The algorithm for the latter class also works for the strong variant of the problem, where the rainbow vertex paths between each vertex pair must be shortest paths. We complement the polynomial-time solvability results for split strongly chordal graphs by showing that, for any fixed $p\ge 3$ both variants of the problem become NP-complete when restricted to split $(S_3,\dots ,S_p)$-free graphs, where $S_q$ denotes the q-sun graph.

给定一个顶点着色图，如果一条路径的所有内部顶点都有不同的颜色，我们就说它是一条彩虹顶点路径。如果每对顶点之间都有一条彩虹顶点路径，则该图是彩虹顶点连接的。在彩虹顶点着色 (RVC)问题中，我们想确定给定图的顶点是否最多可以用k种颜色着色，以便图成为彩虹顶点连接。这个问题已知是NP-即使在非常有限的场景中也能完成，并且很少有已知的有效算法。在这项工作中，我们在排列图、树的幂和分裂强和弦图上给出了 RVC 的多项式时间算法。后一类的算法也适用于该问题的强变体，其中每个顶点对之间的彩虹顶点路径必须是最短路径。我们补充了分裂强弦图的多项式时间可解性结果，表明对于任何固定$磷\ge 3$当限制为分裂时，问题的两个变体都变成NP -complete$({秒}_3,\dots ,{秒}_{磷})$- 自由图，其中 ${秒}_q$表示q -sun 图。