On linear and circular structure of (claw, net)-free graphs

Brandstädt, Andreas; Dragan, Feodor F.

doi:10.1016/s0166-218x(02)00571-1

Cited by 29 publications

(18 citation statements)

References 27 publications

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“…Also note that CN-free graphs are exactly the Hamiltonian-hereditary graphs [13] (was cited in [4]). CN-free graphs turn out to be closely related to AT-free graphs form their structure properties [4]. There are, however, few results about the structure of these graphs [4].…”

Section: Introductionmentioning

confidence: 99%

“…CN-free graphs turn out to be closely related to AT-free graphs form their structure properties [4]. There are, however, few results about the structure of these graphs [4]. In [4] the authors give results on the linear and circular structure of CN-free graphs.…”

Section: Introductionmentioning

confidence: 99%

“…There are, however, few results about the structure of these graphs [4]. In [4] the authors give results on the linear and circular structure of CN-free graphs. AT-free graphs can be generalized in a manner obvious to admit circular structure [4].…”

Section: Introductionmentioning

confidence: 99%

“…In [4] the authors give results on the linear and circular structure of CN-free graphs. AT-free graphs can be generalized in a manner obvious to admit circular structure [4]. CN-free graphs were introduced by Duffus [14].…”

Section: Introductionmentioning

confidence: 99%

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Algorithms for the Recognition of Net-free Graphs and for Computing Maximum Cardinality Matchings in Claw-free Graphs

Talmaciu¹,

Lepin²

2014

SIC

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During the last decades, numerous studies have been undertaken on the classes of net-free graphs, claw-free graphs, and the relationship between them. The notion of weakly decomposition (a partition of the set of vertices in three classes A, B, C such that A induces a connected graph and C is totally adjacent to B and totally non-adjacent to A) and the study of its properties allow us to obtain several important results such as: characterization of cographs, {P4,C4}-free and paw-free graphs. In this article, we give a characterization of net-free graphs, a characterization of claw-free graphs, using weakly decomposition. Also, we give a recognition algorithm for net-free graphs, an algorithm for determining a maximum matching in claw-free graphs, comparable with existing algorithms in terms of complexity, but using weakly decomposition.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Algorithms for the Recognition of Net-free Graphs and for Computing Maximum Cardinality Matchings in Claw-free Graphs

Talmaciu¹,

Lepin²

2014

SIC

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“…G is said to be H-free, if H is not a subgraph of G. If H is a set of graphs, G is H-free if G is H free for all H ∈ H. There is some literature concerning {claw, net}-free graphs, dealing with domination and hamiltonicity problems: It was shown by Damaschke [12] that a connected graph is {claw, net}-free iff each of its connected induced subgraphs has a hamiltonian path. Later, Brandstädt and Dragan [13] studied {claw, net}-free graphs in view of their linear and circular structure. They proved that a connected {claw, net}-free graph either has a doubly dominating induced cycle or a dominating pair, i.e.…”

Section: Introductionmentioning

confidence: 99%

Paired- and induced paired-domination in (E,net)-free graphs

Schaudt¹

2012

Discuss. Math. Graph Theory

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A dominating set of a graph is a vertex subset that any vertex belongs to or is adjacent to. Among the many well-studied variants of domination are the so-called paired-dominating sets. A paired-dominating set is a dominating set whose induced subgraph has a perfect matching. In this paper, we continue their study.We focus on graphs that do not contain the net-graph (obtained by attaching a pendant vertex to each vertex of the triangle) or the E-graph (obtained by attaching a pendant vertex to each vertex of the path on three vertices) as induced subgraphs. This graph class is a natural generalization of (claw,net)-free graphs, which are intensively studied with respect to their nice properties concerning domination and hamiltonicity. We show that any connected (E,net)-free graph has a paired-dominating set that, roughly, contains at most half of the vertices of the graph. This bound is a significant improvement to the known general bounds.Further, we show that any (E,net,C5)-free graph has an induced paireddominating set, that is a paired-dominating set that forms an induced matching, and that such set can be chosen to be a minimum paireddominating sets. We use these results to obtain a new characterization of (E,net,C5)-free graphs in terms of the hereditary existence of induced paired-dominating sets. Finally, we show that the induced matching formed by an induced paired-dominating set in a (E,net,C5)-free graph can be chosen to have at most two times the size of the smallest maximal induced matching possible.

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The Complexity of the Exact Weighted Independent Set Problem

Milanič

Monnot²

2008

Combinatorial Optimization and Theoretical Computer Science

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On linear and circular structure of (claw, net)-free graphs

Cited by 29 publications

References 27 publications

Algorithms for the Recognition of Net-free Graphs and for Computing Maximum Cardinality Matchings in Claw-free Graphs

Algorithms for the Recognition of Net-free Graphs and for Computing Maximum Cardinality Matchings in Claw-free Graphs

Paired- and induced paired-domination in (E,net)-free graphs

The Complexity of the Exact Weighted Independent Set Problem

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