We introduce the Multicolored Graph Realization problem (MGRP). The input to the problem is a colored graph $(G,\varphi)$, i.e., a graph together with a coloring on its vertices. We can associate to each colored graph a cluster graph ($G_\varphi)$ 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 $G_\varphi$. The MGR problem is related to the class of generalized network problems, most of which are NP-hard. 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 $G_\varphi$, among those for treewidth. We look to instances of the problem in which both the number of color classes and the treewidth of $G_\varphi$ are unbounded. We show that MGRP is NP-complete when $G_\varphi$ 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.

中文翻译：

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五彩图实现问题

我们介绍了五彩图实现问题（MGRP）。问题的输入是彩色图形$（G，\ varphi）$，即图形及其顶点上的着色。我们可以将一个聚类图（$ G_ \ varphi）$与每个彩色图相关联，在该聚类图中，将具有相同颜色的所有顶点折叠到一个节点后，我们将删除多个边和自环。当$ S $具有每个颜色类的一个顶点时，一组顶点$ S $就是彩色的。问题是要确定是否存在多色集$ S $，以便在用颜色类别标识$ S $中的每个顶点后，$ G [S] $与$ G_ \ varphi $相符。MGR问题与广义网络问题的类别有关，其中大多数是NP难题的。例如广义的MST问题。MGRP是对多色集团问题的概括，当通过颜色数量进行参数化时，它被认为是W [1] -hard。因此，在树形宽度的那些中，当通过簇图的大小进行参数化并且通过$ G_ \ varphi $上的任何图形参数进行参数化时，MGRP仍然保持W [1] -hard。我们看一下颜色实例的数量和$ G_ \ varphi $的树宽都不受限制的问题的实例。我们证明当$ G_ \ varphi $是弦，双凸二分，完全二分或二维网格时，MGRP是NP完全的。我们的硬度来自于1合3单调SAT问题的适当降低。我们的减少表明，即使在颜色类别中的最大顶点数为3时，问题仍然很棘手。在网格的情况下，硬度也保持带有界度的图。