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.
翻译:我们引入了多色图形现实化问题( MGRP ) 。 当 $S 在每个颜色类中有一个完全的顶点时, 一组的顶点是多色 $( G,\ varphi) 。 问题在于确定多色 3 的 $S 是否是多色 的 。 我们可以将每张彩色的 图形 ( G\ varphi) 和每个彩色类中的彩色 $( G,\ vvarphi) 联系起来。 我们可以将一个聚点与每个彩色的 彩色图形( G+varphi) 相挂钩 。 在结结结结后, 我们的彩色 $O 的双色 美元 。 彩色 彩色 的彩色 3, 我们的彩色 的彩色 中, 我们的彩色 双色 的彩色 的彩色 。