The Multicolored Graph Realization Problem

We introduce the multicolored graph realization problem (MGR). The input to the problem is a colored graph (G,φ), i.e., a graph together with a coloring φ on its vertices. We can associate to each colored graph (G,φ) a cluster graph (Gφ) in which, after collapsing to a node all vertices with the same color, we remove multiple edges and self-loops. A set of vertices S is multicolored when S has exactly one vertex from each color class. The problem is to decide whether there is a multicolored set S such that, after identifying each vertex in S with its color class, G[S] coincides with Gφ. The MGR problem is related to the well known class of generalized network problems, most of which are NP-hard. For example the generalized Minimun Spanning Tree problem. The MGR problem is a generalization of the multicolored clique problem, which is known to be W [1]-hard when parameterized by the number of colors. Thus MGR remains W [1]-hard, when parameterized by the size of the cluster graph. This results implies that the MGR problem is W [1]-hard when parameterized by any graph parameter on Gφ, among those for treewidth. In consequence, we look to instances of the problem in which both the number of color classes and the treewidth of Gφ are unbounded. We consider three natural such graph classes: chordal graphs, convex bipartite graphs and 2-dimensional grid graphs. We show that the MGR problem is NP-complete when Gφ is either chordal, biconvex bipartite, complete bipartite or a 2-dimensional grid. Our hardness results follows from suitable reductions from the 1-in-3 monotone SAT problem. Our reductions show that the problem remains hard even when the maximum number of vertices in a color class is 3. In the case of the grid, the hardness holds also graphs with bounded degree. We complement those results by showing combined parameterizations under which the MGR problem became tractable. ∗Corresponding author Email addresses: diaz@cs.upc.edu (Josep Dı́az), oznur.yasar@khas.edu.tr ( Öznur Yaşar Diner), mjserna@cs.upc.edu (Maria Serna), oriol.serra@upc.edu (Oriol Serra) Preprint submitted to Elsevier March 25, 2021