On claw-free asteroidal triple-free graphs

Hempel, Harald; Kratsch, Dieter

doi:10.1016/s0166-218x(01)00208-6

Cited by 16 publications

(10 citation statements)

References 21 publications

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“…The (independent) domination problem asks for an (independent) dominating set of G with the minimum cardinality, while the independent set problem asks for an independent set of G with the maximum cardinality. To solve these problems we use the structural properties of (claw, net)-free graphs, presented in Section 2, and a few known algorithmic results from [22]. In [22], Hempel and Kratsch gave linear time algorithms for all three problems on the class of claw-free AT-free graphs.…”

Section: Discussionmentioning

confidence: 99%

“…To solve these problems we use the structural properties of (claw, net)-free graphs, presented in Section 2, and a few known algorithmic results from [22]. In [22], Hempel and Kratsch gave linear time algorithms for all three problems on the class of claw-free AT-free graphs. Note that both the domination problem and the independent domination problem are NP-hard in claw-free graphs.…”

Section: Discussionmentioning

confidence: 99%

“…(3) Since G x = G(V \ N[x]) is a claw-free AT-free graph, we can apply a linear time algorithm from [22] to get a minimum independent dominating set D of G x .…”

Section: Discussionmentioning

confidence: 99%

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

Brandstädt

Dragan

2003

Discrete Applied Mathematics

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We prove that every (claw, net)-free graph contains an induced doubly dominating cycle or a dominating pair. Moreover, using LexBFS we present a linear time algorithm which, for a given (claw, net)-free graph, ÿnds either a dominating pair or an induced doubly dominating cycle. We show also how one can use structural properties of (claw, net)-free graphs to solve e ciently the domination, independent domination, and independent set problems on these graphs. ?

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Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

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

Brandstädt

Dragan

2003

Discrete Applied Mathematics

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“…The aim of this section is to prove that 2K 2 -free unit interval graphs are e-positive. Our proof is based on the characterization of 2K 2 -free unit interval graphs due to Hempel and Kratsch [10], who actually gave a characterization of a larger family of graphs. Using their result, we show that 2K 2 -free unit interval graphs can only be either co-triangle-free graphs or generalized bull graphs, which are already known to be e-positive.…”

Section: K 2 -Free Unit Interval Graphsmentioning

confidence: 99%

On the $e$-Positivity of $(claw, 2K_2)$-Free Graphs

Yang

2021

Electron. J. Combin.

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Motivated by Stanley and Stembridge's conjecture about the $e$-positivity of claw-free incomparability graphs, Hamel and her collaborators studied the $e$-positivity of $(claw, H)$-free graphs, where $H$ is a four-vertex graph. In this paper we establish the $e$-positivity of generalized pyramid graphs and $2K_2$-free unit interval graphs, which are two important families of $(claw, 2K_2)$-free graphs. Hence we affirmatively solve one problem proposed by Hamel, Hoáng and Tuero, and another problem considered by Foley, Hoáng and Merkel.

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“…The interval graphs are exactly the chordal AT-free graphs [23]. The complexity of (exact and approximate) diameter computation within AT-free graphs was studied in [9] (see also [14,20]), where the authors emphasize a kind of duality between AT-free graphs and chordal graphs. For instance, on both graph classes, two consecutive executions of LexBFS always yield a vertex whose eccentricity is within one of the diameter -this is the so-called 2-sweep LexBFS algorithm, see Fig.…”

Section: Introductionmentioning

confidence: 99%

Around the diameter of AT-free graphs

Ducoffe

2020

Preprint

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A graph algorithm is truly subquadratic if it runs in O(m b ) time on connected m-edge graphs, for some positive b < 2. Roditty and Vassilevska Williams (STOC'13) proved that under plausible complexity assumptions, there is no truly subquadratic algorithm for computing the diameter of general graphs. In this work, we present positive and negative results on the existence of such algorithms for computing the diameter on some special graph classes. Specifically, three vertices in a graph form an asteroidal triple (AT) if between any two of them there exists a path that avoids the closed neighbourhood of the third one. We call a graph AT-free if it does not contain an AT. We first prove that for all m-edge AT-free graphs, one can compute all the eccentricities in truly subquadratic O(m 3/2 ) time. Then, we extend our study to several subclasses of chordal graphs -all of them generalizing interval graphs in various ways -, as an attempt to understand which of the properties of AT-free graphs, or natural generalizations of the latter, can help in the design of fast algorithms for the diameter problem on broader graph classes. For instance, for all chordal graphs with a dominating shortest path, there is a linear-time algorithm for computing a diametral pair if the diameter is at least four. However, already for split graphs with a dominating edge, under plausible complexity assumptions, there is no truly subquadratic algorithm for deciding whether the diameter is either 2 or 3.

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On claw-free asteroidal triple-free graphs

Cited by 16 publications

References 21 publications

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

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

On the $e$-Positivity of $(claw, 2K_2)$-Free Graphs

Around the diameter of AT-free graphs

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