The complexity of determining the vertex-rainbow index of graphs

Abstract

The concept of the [Formula: see text]-rainbow index of a network comes from the communication of information between agencies of government, which was introduced by Chartrand et al. in 2010. As a natural counterpart of the [Formula: see text]-rainbow index, the concept of [Formula: see text]-vertex-rainbow index was also introduced. For a graph [Formula: see text] and a set [Formula: see text] of at least two vertices, an [Formula: see text]-Steiner tree or a Steiner tree connecting [Formula: see text] (or simply, an [Formula: see text]-tree) is such a subgraph [Formula: see text] of [Formula: see text] that is a tree with [Formula: see text]. For [Formula: see text] and [Formula: see text], an [Formula: see text]-Steiner tree [Formula: see text] is said to be a vertex-rainbow [Formula: see text]-tree if the vertices of [Formula: see text] have distinct colors. For a fixed integer [Formula: see text] with [Formula: see text], the vertex-coloring [Formula: see text] of [Formula: see text] is called a [Formula: see text]-vertex-rainbow coloring if for every [Formula: see text]-subset [Formula: see text] of [Formula: see text] there exists a vertex-rainbow [Formula: see text]-tree. In this case, [Formula: see text] is called vertex-rainbow [Formula: see text]-tree-connected. The minimum number of colors that are needed in a [Formula: see text]-vertex-rainbow coloring of [Formula: see text] is called the [Formula: see text]-vertex-rainbow index of [Formula: see text], denoted by [Formula: see text]. In this paper, we study the complexity of determining the [Formula: see text]-vertex-rainbow index of a graph and prove that computing [Formula: see text] is [Formula: see text]-Hard. Moreover, we show that it is [Formula: see text]-Complete to decide whether [Formula: see text]. We also prove that the following problem is [Formula: see text]-Complete: Given a vertex-colored graph [Formula: see text], check whether the given coloring makes [Formula: see text] vertex-rainbow [Formula: see text]-tree-connected.