Parameterized algorithms for recognizing monopolar and 2-subcolorable graphs

Kanj, Iyad A.; Komusiewicz, Christian; Sorge, Manuel; Leeuwen, Erik Jan van

doi:10.1016/j.jcss.2017.08.002

Cited by 3 publications

(8 citation statements)

References 33 publications

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“…Thus, we consider the number k of vertices in the graph G [B], even for the broader (Π A , Π B )-Recognition problem. We previously proved a general fixed-parameter tractability result in this case [22]. We observe a very general kernelization result: We obtain a better bound in terms of the number of vertices for Cluster-Π ∆ -Partition, the restriction of Cluster-Π-Partition to the case when all graphs containing a vertex of degree at least ∆ + 1 are forbidden induced subgraphs of Π.…”

Section: Our Resultsmentioning

confidence: 61%

“…Similar effects are visible in the aforementioned cochromatic number and rectangle stabbing number problems [21,24]. The contribution of the previous works [21,22,24] was to bound the depth of this process by some function of the parameter, leading to fixed-parameter algorithms. However, such a bound does not provide an answer to the question of which vertices trigger avalanches and their continued rolling, and whether the number of such vertices can somehow be limited.…”

Section: Introductionmentioning

confidence: 68%

“…As we have seen in this paper, the pushing process is not only useful for finding efficient algorithms for (Π A , Π B )-Recognition as demonstrated by Kanj et al [22], but can also be used to classify when or when not such problems admit polynomial kernels. Herein, we focused on the well-motivated case when Π A is the set of cluster graphs on the first level above triviality; when Π A is characterized by forbidden induced subgraphs of order at least three.…”

Section: Discussionmentioning

confidence: 90%

“…Many such (Π A , Π B )-Recognition problems were shown fixed-parameter tractable by Kanj et al [22], for example when Π A is the class of graphs that is a disjoint union of k cliques, using parameter k. The central algorithmic idea that was employed in [22] is the pushing process. The algorithm empties the input graph, and adds vertices back one by one while maintaining a valid partition.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Solving Partition Problems Almost Always Requires Pushing Many Vertices Around

Kanj¹,

Komusiewicz²,

Sorge³

et al. 2020

SIAM J. Discrete Math.

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A fundamental graph problem is to recognize whether the vertex set of a graph G can be bipartitioned into sets A and B such that G [A] and G [B] satisfy properties ΠA and ΠB, respectively. This so-called (ΠA, ΠB)-Recognition problem generalizes amongst others the recognition of 3-colorable, bipartite, split, and monopolar graphs. A powerful algorithmic technique that can be used to obtain fixed-parameter algorithms for many cases of (ΠA, ΠB)-Recognition, as well as several other problems, is the pushing process. For bipartition problems, the process starts with an "almost correct" bipartition (A , B ), and pushes appropriate vertices from A to B and vice versa to eventually arrive at a correct bipartition.In this paper, we study whether (ΠA, ΠB)-Recognition problems for which the pushing process yields fixed-parameter algorithms also admit polynomial problem kernels. In our study, we focus on the first level above triviality, where ΠA is the set of P3-free graphs (disjoint unions of cliques, or cluster graphs), the parameter is the number of clusters in the cluster graph G[A], and ΠB is characterized by a set H of connected forbidden induced subgraphs. We prove that, under the assumption that NP ⊆ coNP/poly, (ΠA, ΠB)-Recognition admits a polynomial kernel if and only if H contains a graph with at most 2 vertices. In both the kernelization and the lower bound results, we make crucial use of the pushing process. Theorem 1.3. Cluster-Π ∆ -Partition parameterized by k, the maximum size of B, has an O((∆ 2 + 1) · k 2 )-vertex kernel. PreliminariesGraphs We follow standard graph-theoretic notation [11]. Let G be a graph. By V (G) and E(G) we denote the vertex-set and the edge-set of G, respectively. Throughout the paper, we use n := |V (G)| to denote the number of vertices in G and m := |E(G)| to denote its number of edges. We also say that G is of order |V (G)|. We assume n = O(m) since isolated vertices can be safely removed in the problems that we consider. For X ⊆ V (G), G[X] = (X, {e | e ∈ E(G) ∩ X} denotes the subgraph of G induced by X. For a vertex v ∈ G, N (v) = {u | {u, v} ∈ E(G)} and N [v] = N (v) ∪ {v} denote the open neighborhood and the closed neighborhood of v, respectively. For X ⊆ V (G), we define N (X) := ( v∈X N (v)) \ X and N [X] := v∈X N [v], and for a family X of subsets X ⊆ V (G), we define N (X ) := ( X∈X N (X)) \ ( X∈X X) and N [X ] := X∈X N [X].

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Section: Our Resultsmentioning

confidence: 61%

Section: Introductionmentioning

confidence: 68%

Section: Discussionmentioning

confidence: 90%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Solving Partition Problems Almost Always Requires Pushing Many Vertices Around

Kanj¹,

Komusiewicz²,

Sorge³

et al. 2020

SIAM J. Discrete Math.

Self Cite

View full text Add to dashboard Cite

show abstract

“…In this context, the graph partitioning problem is NP-complete [24], and there are available strategies based on spectral [25] (eigenproblem in [26]), combinatorial [27], geometric [28] and multi-level [29] heuristics. Partitioning of the graph vertices leads to recognition the of 2-subcolorable [30], bipartite [31], cluster [32], dominable [33], monopolar [34], r-partite [35], split [36], unipolar [37], trapezoid [38] and graphical algorithms (etc.) working efficiently with special classes of graphs that have been devised (for monopolar and 2-subcolorable in [30]; for unipolar and generalized split in [39]; for partitioning a big graph into k sub-graphs in [40,41]; for graph that does not contain an induced subgraph, a claw in [42]).…”

Section: Related Researchmentioning

confidence: 99%

Figures of Graph Partitioning by Counting, Sequence and Layer Matrices

2021

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A series of counting, sequence and layer matrices are considered precursors of classifiers capable of providing the partitions of the vertices of graphs. Classifiers are given to provide different degrees of distinctiveness for the vertices of the graphs. Any partition can be represented with colors. Following this fundamental idea, it was proposed to color the graphs according to the partitions of the graph vertices. Two alternative cases were identified: when the order of the sets in the partition is relevant (the sets are distinguished by their positions) and when the order of the sets in the partition is not relevant (the sets are not distinguished by their positions). The two isomers of C28 fullerenes were colored to test the ability of classifiers to generate different partitions and colorings, thereby providing a useful visual tool for scientists working on the functionalization of various highly symmetrical chemical structures.

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Monopolar graphs: Complexity of computing classical graph parameters

Barbato

Bezzi

2021

Discrete Applied Mathematics

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Parameterized algorithms for recognizing monopolar and 2-subcolorable graphs

Cited by 3 publications

References 33 publications

Solving Partition Problems Almost Always Requires Pushing Many Vertices Around

Solving Partition Problems Almost Always Requires Pushing Many Vertices Around

Figures of Graph Partitioning by Counting, Sequence and Layer Matrices

Monopolar graphs: Complexity of computing classical graph parameters

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