Browsing by Author "Diaz, Josep"

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Citation Count: 0
The multicolored graph realization problem
(Elsevier, 2024) Diaz, Josep; Diner, Oznur Yasar; Serna, Maria; Serra, Oriol
We introduce the multicolored graph realization problem (MGR). The input to this problem is a colored graph ( G , phi ), i.e., a graph G together with a coloring phi on its vertices. We associate each colored graph ( G , phi ) with a cluster graph ( G phi ) 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 G phi . 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 G phi , 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 G phi 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 G phi 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. (c) 2022 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
Citation Count: 6
On list k-coloring convex bipartite graphs
(Springer Nature, 2021) Diaz, Josep; Diner, Öznur Yaşar; Serna, Maria; Oriol, Serra
List k-Coloring (Lik-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 Lik-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 et al. (SIAM J Discrete Math 28(4):1675–1685, 2014) can be applied to the problem. We provide a dynamic programming algorithm to solve the Lik-Col in the class of convex bipartite graphs. Finally, we show how our algorithm can be modified to solve the more general LiH-Col problem on convex bipartite graphs.
Citation Count: 0
On minimum vertex bisection of random d-regular graphs
(Academic Press inc Elsevier Science, 2024) Diaz, Josep; Diner, Oznur Yasar; Serna, Maria; Serra, Oriol
Minimum vertex bisection is a graph partitioning problem in which the aim is to find a partition of the vertices into two equal parts that minimizes the number of vertices in one partition set that have a neighbor in the other set. In this work we are interested in providing asymptotically almost surely upper bounds on the minimum vertex bisection of random d -regular graphs, for constant values of d . Our approach is based on analyzing a greedy algorithm by using the differential equation method. In this way, we obtain the first known non -trivial upper bounds for the vertex bisection number in random regular graphs. The numerical approximations of these theoretical bounds are compared with the emprical ones, and with the lower bounds from Kolesnik and Wormald (2014) [30]. (c) 2024 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons .org /licenses /by /4 .0/).