The multicolored graph realization problem

We introduce the multicolored graph realization problem (MGR). The input to this problem is a colored graph (G, φ), i.e., a graph G together with a coloring φ on its vertices. We associate each colored graph (G, φ) with a cluster graph (Gφ ) in which, after collapsing all vertices with the same color to a node, 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 MGR problem is to decide whether there is a multicolored set S so 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, like the generalized Minimum Spanning Tree problem. The MGR 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. These results imply that the MGR problem is W [1]-hard when parameterized by any graph parameter on Gφ , among which lies treewidth. Consequently, we look at the 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 MGR is NP-complete when Gφ is either chordal, biconvex bipartite, complete bipartite or a 2-dimensional grid. 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 even for graphs with bounded degree. We provide a complexity dichotomy with respect to cluster size .J. Díaz and M. Serna are partially supported by funds from the Spanish Agencia Estatal de Investigación under grant PID2020-112581GB-C21 (MOTION), and from AGAUR under grant 2017-SGR-786 (ALBCOM). Ö. Y. Diner is partially supported by the Scientific and Technological Research Council Tübitak under project BIDEB 2219-1059B191802095 and by Kadir Has University under project 2018-BAP-08. O. Serra is supported by the Spanish Agencia Estatal de Investigación under grant PID2020-113082GB-I00.Peer ReviewedPostprint (published version

On list k-coloring convex bipartite graphs

List k–Coloring (LI k-COL) is the decision problem asking if a given graph admits a proper coloring compatible with a given list assignment to its vertices with colors in {1,2,..., k}. The problem is known to be NP-hard even for k = 3 within the class of 3–regular planar bipartite graphs and for k = 4 within the class of chordal bipartite graphs. In 2015 Huang, Johnson and Paulusma asked for the complexity of LI 3-COL in the class of chordal bipartite graphs. In this paper, we give a partial answer to this question by showing that LI k-COL is polynomial in the class of convex bipartite graphs. We show first that biconvex bipartite graphs admit a multichain ordering, extending the classes of graphs where a polynomial algorithm of Enright, Stewart and Tardos (2014) can be applied to the problem. We provide a dynamic programming algorithm to solve the LI k-COL in the class of convex bipartite graphs. Finally, we show how our algorithm can be modified to solve the more general LI H-COL problem on convex bipartite graphs.J. Díaz and M. Serna are partially supported by funds from MINECO and EU FEDER under grant TIN 2017-86727-C2-1-R AGAUR project ALBCOM 2017- SGR-786. O. Y. Diner is partially supported by the Scientific and Technological Research Council Tubitak under project BIDEB 2219-1059B191802095 and by Kadir Has University under project 2018-BAP-08. O. Serra is supported by the Spanish Ministry of Science under project MTM2017-82166-P.Peer ReviewedPostprint (author's final draft