Decomposition of graphs

Tyshkevich, R. I.; Chernyak, A. A.

doi:10.1007/bf01072106

Cited by 31 publications

(12 citation statements)

References 8 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…However, polar graphs stand in our way. A graph G = (V, E) has a polar partition if V can be partitioned into sets A and B such that G[A] is a cluster graph and G[B] is the complement of a cluster graph (a co-cluster graph) [31]. We have the following proposition: Proposition 9.3.…”

Section: Hardness Resultsmentioning

confidence: 99%

Parameterized algorithms for recognizing monopolar and 2-subcolorable graphs

Kanj

Komusiewicz

Sorge

et al. 2018

Journal of Computer and System Sciences

View full text Add to dashboard Cite

A graph G is a (ΠA, ΠB)-graph if V (G) can be bipartitioned into A and B such that G[A] satisfies property ΠA and G [B] satisfies property ΠB. The (ΠA, ΠB)-Recognition problem is to recognize whether a given graph is a (ΠA, ΠB)-graph. There are many (ΠA, ΠB)-Recognition problems, including the recognition problems for bipartite, split, and unipolar graphs. We present efficient algorithms for many cases of (ΠA, ΠB)-Recognition based on a technique which we dub inductive recognition. In particular, we give fixed-parameter algorithms for two NP-hard (ΠA, ΠB)-Recognition problems, Monopolar Recognition and 2-Subcoloring, parameterized by the number of maximal cliques in G [A]. We complement our algorithmic results with several hardness results for (ΠA, ΠB)-Recognition.

show abstract

Section: Hardness Resultsmentioning

confidence: 99%

Parameterized algorithms for recognizing monopolar and 2-subcolorable graphs

Kanj

Komusiewicz

Sorge

et al. 2018

Journal of Computer and System Sciences

View full text Add to dashboard Cite

show abstract

“…Also, in it was shown that if a graph admits such a partition, then the black vertices form an induced matching of maximum size. Notice that a graph is called polar if its vertex set can be partitioned into a subset

scriptK

of disjoint cliques and a subset

scriptI

of independent sets with complete links between them . It follows that a graph G has a dominating induced matching if and only if G is a polar graph in which all cliques of

scriptK

have size 2 and

scriptI

consists of exactly one independent set.…”

Section: Introductionmentioning

confidence: 99%

Dominating induced matchings in graphs containing no long claw

Hertz

Ries

Zamaraev

et al. 2017

Journal of Graph Theory

View full text Add to dashboard Cite

An induced matching M in a graph G is dominating if every edge not in M shares exactly one vertex with an edge in M. The dominating induced matching problem (also known as efficient edge domination) asks whether a graph G contains a dominating induced matching. This problem is generally NP‐complete, but polynomial‐time solvable for graphs with some special properties. In particular, it is solvable in polynomial time for claw‐free graphs. In the present article, we provide a polynomial‐time algorithm to solve the dominating induced matching problem for graphs containing no long claw, that is, no induced subgraph obtained from the claw by subdividing each of its edges exactly once.

show abstract

“…Polar and monopolar graphs were introduced in as common generalizations of bipartite and split graphs. ( Split graphs are the graphs which can be partitioned into an independent set and a clique (cf.…”

Section: Introductionmentioning

confidence: 99%

On the Polarity and Monopolarity of Graphs

Churchley

Huang

2013

Journal of Graph Theory

View full text Add to dashboard Cite

Abstract:Polarity and monopolarity are properties of graphs defined in terms of the existence of certain vertex partitions; graphs with polarity and monopolarity are respectively called polar and monopolar graphs. These two properties commonly generalize bipartite and split graphs, but are hard to recognize in general. In this article we identify two classes of graphs, triangle-free graphs and claw-free graphs, restricting to which provide novel impact on the complexity of the recognition problems. More precisely, we prove that recognizing polarity or monopolarity remains NP-complete for triangle-free graphs. We also show that for claw-free graphs the former is NP-complete and the latter is polynomial time solvable. This is in sharp contrast to a recent result that both polarity and monopolarity can be recognized in linear time for line graphs. Our proofs for the NP-completeness are simple reductions. The polynomial time algorithm for recognizing the monopolarity of claw-free graphs uses a subroutine similar to the well-known breadth-first search algorithm and is based on a new structural characterization of monopolar claw-free graphs, a generalization of one for monopolar line graphs obtained earlier. C 2013 Wiley Periodicals, Inc. J.

show abstract

Decomposition of graphs

Cited by 31 publications

References 8 publications

Parameterized algorithms for recognizing monopolar and 2-subcolorable graphs

Parameterized algorithms for recognizing monopolar and 2-subcolorable graphs

Dominating induced matchings in graphs containing no long claw

On the Polarity and Monopolarity of Graphs

Contact Info

Product

Resources

About