On List k-Coloring Convex Bipartite Graphs

Dı́az, Josep; Diner, Öznur Yaşar; Serna, María; Serra, Oriol

doi:10.1007/978-3-030-63072-0_2

Cited by 4 publications

(2 citation statements)

References 30 publications

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“…Further Algorithmic Consequences. Theorems 1 and 2, combined with the result of [28], also generalize a result of Díaz et al [17] for List k-Colouring on convex graphs to circular convex and (t, ∆)-tree convex graphs (t ≥ 1, ∆ ≥ 3).…”

Section: Mim-widthsupporting

confidence: 71%

Solving Problems on Generalized Convex Graphs via Mim-Width

Bonomo

Brettell

Munaro

et al. 2021

Lecture Notes in Computer Science

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A bipartite graph G = (A, B, E) is H-convex, for some family of graphs H, if there exists a graph H ∈ H with V (H) = A such that the set of neighbours in A of each b ∈ B induces a connected subgraph of H. Many NP-complete problems become polynomial-time solvable for H-convex graphs when H is the set of paths. In this case, the class of H-convex graphs is known as the class of convex graphs. The underlying reason is that this class has bounded mim-width. We extend the latter result to families of H-convex graphs where (i) H is the set of cycles, or (ii) H is the set of trees with bounded maximum degree and a bounded number of vertices of degree at least 3. As a consequence, we can reprove and strengthen a large number of results on generalized convex graphs known in the literature. To complement result (ii), we show that the mim-width of H-convex graphs is unbounded if H is the set of trees with arbitrarily large maximum degree or an arbitrarily large number of vertices of degree at least 3. In this way we are able to determine complexity dichotomies for the aforementioned graph problems. Afterwards we perform a more refined width-parameter analysis, which shows even more clearly which width parameters are bounded for classes of H-convex graphs.Brettell and Paulusma received support from the Leverhulme Trust (RPG-2016-258). Bonomo received support from UBACyT (20020170100495BA and 20020160100095BA). Brettell also received support from a Rutherford Foundation Postdoctoral Fellowship, administered by the Royal Society Te Apārangi.

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Section: Mim-widthsupporting

confidence: 71%

Solving Problems on Generalized Convex Graphs via Mim-Width

Bonomo

Brettell

Munaro

et al. 2021

Lecture Notes in Computer Science

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“…In particular, we consider the subclasses of convex bipartite graphs and biconvex bipartite graphs which have been used as a benchmark for complexity of homomorphism problems, see e.g. [15,16,17]. We show that the MGR problem is NP-complete for colored graphs whose cluster graph is biconvex bipartite (see section 3).…”

Section: Introductionmentioning

confidence: 99%

The Multicolored Graph Realization Problem

Dı́az¹,

Diner²,

Serna³

et al. 2021

Preprint

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We introduce the multicolored graph realization problem (MGR). The input to the problem is a colored graph (G, ϕ), i.e., a graph together with a coloring ϕ on its vertices. We can associate to each colored graph (G, ϕ) a cluster graph (G ϕ ) 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 ϕ .The MGR problem is related to the well known class of generalized network problems, most of which are NP-hard. For example the generalized Minimun Spanning Tree problem. The MGR problem 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. This results implies that the MGR problem is W [1]-hard when parameterized by any graph parameter on G ϕ , among those for treewidth. In consequence, we look to instances of the problem in which both the number of color classes and the treewidth of G ϕ are unbounded. We consider three natural such graph classes: chordal graphs, convex bipartite graphs and 2-dimensional grid graphs. We show that the MGR problem is NP-complete when G ϕ 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.

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