Complexity of rainbow vertex connectivity problems for restricted graph classes

摘要

A path in a vertex-colored graph G is vertex rainbow if all of its internal vertices have a distinct color. The graph G is said to be rainbow vertex connected if there is a vertex rainbow path between every pair of its vertices. Similarly, the graph G is strongly rainbow vertex connected if there is a shortest path which is vertex rainbow between every pair of its vertices. We consider the complexity of deciding if a given vertex-colored graph is rainbow or strongly rainbow vertex connected. We call these problems RAINBOW VERTEX CONNECTIVITY and STRONG RAINBOW VERTEX CONNECTIVITY, respectively. We prove both problems remain NP-complete on very restricted graph classes including bipartite planar graphs of maximum degree 3, interval graphs, and k-regular graphs for k >= 3. We settle precisely the complexity of both problems from the viewpoint of two width parameters: pathwidth and tree-depth. More precisely, we show both problems remain NP-complete for bounded pathwidth graphs, while being fixed-parameter tractable parameterized by tree-depth. Moreover, we show both problems are solvable in polynomial time for block graphs, while STRONG RAINBOW VERTEX CONNECTIVITY is tractable for cactus graphs and split graphs. (C) 2016 Elsevier B.V. All rights reserved.