
A Complexity Dichotomy for Critical Values of the bChromatic Number of Graphs
A bcoloring of a graph G is a proper coloring of its vertices such that...
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The Maximum Colorful Arborescence problem parameterized by the structure of its color hierarchy graph
Let G=(V,A) be a vertexcolored arcweighted directed acyclic graph (DAG...
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The Satisfactory Partition Problem
The Satisfactory Partition problem consists in deciding if the set of ve...
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Finding Small Weight Isomorphisms with Additional Constraints is FixedParameter Tractable
Lubiw showed that several variants of Graph Isomorphism are NPcomplete,...
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Parameterized (Approximate) Defective Coloring
In Defective Coloring we are given a graph G = (V, E) and two integers χ...
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Recognizing kClique Extendible Orderings
A graph is kcliqueextendible if there is an ordering of the vertices s...
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How to navigate through obstacles?
Given a set of obstacles and two points, is there a path between the two...
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The Multicolored Graph Realization Problem
We introduce the Multicolored Graph Realization problem (MGRP). The input to the problem is a colored graph (G,φ), i.e., a graph together with a coloring on its vertices. We can associate to each colored graph a cluster graph (G_φ) in which, after collapsing to a node all vertices with the same color, we remove multiple edges and selfloops. 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 G_φ. The MGR problem is related to the class of generalized network problems, most of which are NPhard. For example the generalized MST problem. MGRP is a generalization of the Multicolored Clique Problem, which is known to be W[1]hard when parameterized by the number of colors. Thus MGRP remains W[1]hard, when parameterized by the size of the cluster graph and when parameterized by any graph parameter on G_φ, among those for treewidth. We look to instances of the problem in which both the number of color classes and the treewidth of G_φ are unbounded. We show that MGRP is NPcomplete when G_φ is either chordal, biconvex bipartite, complete bipartite or a 2dimensional grid. Our hardness results follows from suitable reductions from the 1in3 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.
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