Further hardness results on the rainbow vertex-connection number of graphs

A vertex-colored graph $G$ is {\it rainbow vertex-connected} if any pair of vertices in $G$ are connected by a path whose internal vertices have distinct colors, which was introduced by Krivelevich and Yuster. The {\it rainbow vertex-connection number} of a connected graph $G$, denoted by $rvc(G)$, is the smallest number of colors that are needed in order to make $G$ rainbow vertex-connected. In a previous paper we showed that it is NP-Complete to decide whether a given graph $G$ has $rvc(G)=2$. In this paper we show that for every integer $k\geq 2$, deciding whether $rvc(G)\leq k$ is NP-Hard. We also show that for any fixed integer $k\geq 2$, this problem belongs to NP-class, and so it becomes NP-Complete.

💡 Research Summary

The paper investigates the computational complexity of the rainbow vertex‑connection number (rvc) of a graph, extending previous results that were limited to the case rvc(G)=2. The rainbow vertex‑connection number of a connected graph G is the smallest number of colors needed to color the vertices so that for every pair of distinct vertices u and v there exists a u‑v path whose internal vertices all receive distinct colors. Basic observations are recalled: rvc(G)=0 iff G is a complete graph, rvc(G)=|V(G)|−2 iff G is a tree, and rvc(G)≥diam(G)−1, with equality for diameters 1 or 2. Prior work had shown that deciding whether rvc(G)=2 is NP‑Complete, but the hardness for arbitrary fixed k≥2 remained open.

The authors introduce an intermediate decision problem called the k‑subset rainbow vertex‑connection problem. An instance consists of a graph G, a set P of ordered vertex pairs, and an integer k. The question is whether there exists a vertex‑coloring of G with at most k colors such that every pair (u,v)∈P is rainbow‑vertex‑connected (i.e., there is a u‑v path whose internal vertices have pairwise distinct colors).

To prove NP‑hardness of this intermediate problem, they give a polynomial‑time reduction from the classic k‑vertex‑coloring problem, which is known to be NP‑Hard for k≥3. Given an instance (G,k) of k‑coloring, they construct a new graph G′ by adding, for each original vertex v, a pendant vertex x_v and the edge (v,x_v). The set P is defined as {(x_u,x_v) | (u,v)∈E(G)}. They show that G is k‑colorable iff G′ admits a k‑coloring in which every pair of vertices from P is rainbow‑vertex‑connected. The forward direction uses the original proper coloring on both original and pendant vertices; the reverse direction extracts a proper coloring of G from any feasible coloring of G′, because any rainbow‑vertex‑connected path between x_u and x_v must pass through u and v, forcing them to have distinct colors. Consequently, the k‑subset rainbow vertex‑connection problem is NP‑Hard for any fixed k≥3.

The second major step is to relate the k‑subset problem to the original decision problem “rvc(G)≤k”. The authors construct, for each k, a graph G_k that embeds the original instance (G,P) in such a way that G_k can be rainbow‑vertex‑connected with k colors if and only if the original instance is a yes‑instance of the k‑subset problem. For k=2 and k=3 they give explicit constructions G_2 and G_3. In G_2, each original vertex v is represented by a vertex v_2 together with auxiliary vertices and edges that enforce the following: if (v_i,v_j)∈P then any rainbow‑vertex‑connected path between the corresponding vertices in G_2 must be long (≥4), while if the pair is not in P there exists a short (length 3) path. The coloring of G_2 with two colors mirrors a feasible solution of the 2‑subset problem. G_3 is built analogously with three layers, ensuring a length gap of 5 versus 4.

For general k≥4 the authors describe an inductive construction. Assuming G_{k‑2} has already been built, each vertex v_{i,k‑2} of G_{k‑2} is split into two copies v_{i,k‑2}^{(1)} and v_{i,k‑2}^{(2)}; all incident edges are duplicated accordingly. A new layer of vertices v_{i,k} (one per original vertex) is added and connected to both copies. The resulting graph G_k contains a subgraph H_k induced by the vertices {v_{i,k}} that is isomorphic to the original graph G. The construction guarantees two crucial properties: (1) for any pair (v_i,v_j)∈P, the shortest path between v_{i,k} and v_{j,k} that avoids edges of H_k has length at least k+2; (2) for any pair not in P, the shortest such path has length exactly k+1. These properties are proved by induction on k.

With these properties, the authors prove a bidirectional equivalence: G is k‑subset rainbow‑vertex‑connected iff G_k is k‑rainbow‑vertex‑connected. The forward direction uses a coloring of G that satisfies the subset condition to color G_k layer by layer, assigning new colors k‑1 and k to the newly introduced split vertices and preserving the original colors on the H_k layer. The backward direction extracts a coloring of the original graph from any feasible coloring of G_k, because any rainbow‑vertex‑connected path for a pair in P must stay inside H_k (otherwise it would be too long), thereby inducing a rainbow‑vertex‑connected path in the original graph.

Since the k‑subset problem is NP‑Hard and the reduction to the rvc decision problem is polynomial, the latter is also NP‑Hard for every fixed integer k≥2. Moreover, a certificate consisting of a k‑coloring can be verified in polynomial time, placing the problem in NP. Consequently, for each fixed k≥2 the problem “given a graph G, decide whether rvc(G)≤k” is NP‑Complete.

The paper’s contributions are threefold: (1) it introduces the k‑subset rainbow vertex‑connection problem and shows its NP‑Hardness via a clean reduction from k‑vertex‑coloring; (2) it develops a systematic, inductive graph transformation that establishes a polynomial‑time equivalence between the subset problem and the rvc decision problem for any k; (3) it finally resolves the open question of the complexity of rvc(G)≤k for arbitrary fixed k, extending the known NP‑Completeness result for k=2 to all k≥2. The techniques presented provide a versatile framework for analyzing parameterized rainbow‑connection problems and may be adapted to other variants of graph connectivity that involve coloring constraints.