Partitions of graphs into one or two independent sets and cliques

Brandstädt, Andreas

doi:10.1016/0012-365x(94)00296-u

Cited by 85 publications

(105 citation statements)

References 2 publications

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“…The main results of the paper are that using different definitions of split graphs [4,5,7,8,9,10,21,22,23,26,28,30,67,82,86,87,88] we propose quick algorithms for the recognition and extremal reconstruction of split sequences among integer, regular [17,45] and graphic [43,45,48] sequences. …”

Section: Definition 3 (Gyárfás [30]) a Graph G Is Called (L M)-boundmentioning

confidence: 99%

Recognition of split-graphic sequences

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Pirzada

Iványi

2014

Acta Universitatis Sapientiae, Informatica

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Section: Definition 3 (Gyárfás [30]) a Graph G Is Called (L M)-boundmentioning

confidence: 99%

Recognition of split-graphic sequences

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Pirzada

Iványi

2014

Acta Universitatis Sapientiae, Informatica

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“…The Cochromatic Number problem is NP-complete by the result of Wagner [32] who proved it to be NP-complete even on permutation graphs. Brandstädt [2] showed that we can recognize in polynomial time whether the vertex set of a given undirected graph can be partitioned into one or two independent sets and one or two cliques. However, it remains NP-complete to check whether we can partition the given graph into κ independent sets or cliques if either κ ≥ 3 or ≥ 3.…”

Section: Cochromatic Numbermentioning

confidence: 99%

Fixed-Parameter Algorithms for Cochromatic Number and Disjoint Rectangle Stabbing

Heggernes

Kratsch²,

Lokshtanov

et al. 2010

Lecture Notes in Computer Science

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Given a permutation π of {1, . . . , n} and a positive integer k, we give an algorithm with running time 2 O(k 2 log k) n O(1) that decides whether π can be partitioned into at most k increasing or decreasing subsequences. Thus we resolve affirmatively the open question of whether the problem is fixed parameter tractable. This NP-complete problem is equivalent to deciding whether the cochromatic number, partitioning into the minimum number of cliques or independent sets, of a given permutation graph on n vertices is at most k. In fact, we give a more general result: within the mentioned running time, one can decide whether the cochromatic number of a given perfect graph on n vertices is at most k.To obtain our result we use a combination of two well-known techniques within parameterized algorithms, namely greedy localization and iterative compression. We further demonstrate the power of this combination by giving a 2 O(k 2 log k) n log n time algorithm for deciding whether a given set of n non-overlapping axis-parallel rectangles can be stabbed by at most k of a given set of horizontal and vertical lines. Whether such an algorithm exists was mentioned as an open question in several papers.

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“…The cochromatic number z(G) of G is the minimum number r of sets (each of which is either a clique or an independent set) that V (G) can be partitioned into. Brandstädt showed that determining if V (G) can be partitioned into r ≤ 2 independent sets and l ≤ 2 cliques is polynomial-time solvable [3]. However, the problem is NP-complete if either r ≥ 3 or l ≥ 3 as it includes 3-Coloring as a special case.…”

Section: Introductionmentioning

confidence: 99%

Parameterized Algorithms for (r,l)-Partization

Krithika¹,

Narayanaswamy²

2013

JGAA

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We consider the (r, l)-Partization problem of finding a set of at most k vertices whose deletion results in a graph that can be partitioned into r independent sets and l cliques. Restricted to perfect graphs and split graphs, we describe sequacious fixed-parameter tractability results for (r, 0)-Partization, parameterized by k and r. For (r, l)-Partization where r + l = 2, we describe an O * (2 k ) algorithm for perfect graphs. We then study the parameterized complexity hardness of a generalization of the Above Guarantee Vertex Cover by a reduction from (r, l)-Partization.

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Partitions of graphs into one or two independent sets and cliques

Cited by 85 publications

References 2 publications

Recognition of split-graphic sequences

Recognition of split-graphic sequences

Fixed-Parameter Algorithms for Cochromatic Number and Disjoint Rectangle Stabbing

Parameterized Algorithms for (r,l)-Partization

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