Local Clique Covering of Claw-Free Graphs

Javadi, Ramin; Maleki, Zeinab; Omoomi, Behnaz

doi:10.1002/jgt.21864

Cited by 12 publications

(12 citation statements)

References 23 publications

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“…[9].) Concerning local clique cover number, R. Javadi et al showed in [12] that if G is a claw-free graph then lcc(G) ≤ c ∆(G) log(∆(G)) , for a constant c. In this section, we are going to prove that Conjecture 3 does hold for claw-free graphs.…”

Section: Claw-free Graphsmentioning

confidence: 88%

Clique coverings and claw-free graphs

Bujtás

Davoodi

Győri

et al. 2020

European Journal of Combinatorics

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Section: Claw-free Graphsmentioning

confidence: 88%

Clique coverings and claw-free graphs

Bujtás

Davoodi

Győri

et al. 2020

European Journal of Combinatorics

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“…Our work is organized as follows. Section 2 contains our main results on

ecc (G)

, where we improve upon the previously best known bound of

ecc (G) \leq c n^{4 ∕ 3} \log^{1 ∕ 3} n

α (G) = 2

, given by Javadi, Maleki, and Omoomi [8]. We also show how our results can be used to obtain even better bounds for fractional edge clique covers.…”

Section: Introductionmentioning

confidence: 66%

Edge clique covers in graphs with independence number two

Charbit

Hahn

Kamiński

et al. 2021

Journal of Graph Theory

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The edge clique cover number ecc ( G ) of a graph G is the size of the smallest collection of complete subgraphs whose union covers all edges of G. Chen, Jacobson, Kézdy, Lehel, Scheinerman, and Wang conjectured in 2000 that if G is claw‐free, then ecc ( G ) is bounded above by its order (denoted n). Recently, Javadi and Hajebi verified this conjecture for claw‐free graphs with an independence number at least three. We study the edge clique cover number of graphs with independence number two, which are necessarily claw‐free. We give the first known proof of a linear bound in n for ecc ( G ) for such graphs, improving upon the bou nd of O ( n 4 ∕ 3 log 1 ∕ 3 n ) due to Javadi, Maleki, and Omoomi. More precisely we prove that ecc ( G ) is at most the minimum of n + δ ( G ) and 2 n − normalΩ ( n log n ), where δ ( G ) is the minimum degree of G. In the fractional version of the problem, we improve these upper bounds to 3 2 n. We also verify the conjecture for some specific subfamilies, for example, when the edge packing number with respect to cliques (a lower bound for ecc ( G )) equals n, and when G contains no induced subgraph isomorphic to H where H is any fixed graph of order 4.

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“…Notice that this problem is related to the local biclique cover problem where one is only interested in minimizing the number of distinct bicliques covering each node without minimizing the total number of bicliques [16]. Clearly, if a graph satisfies the ( f L , f R , k)-biclique cover condition it also satisfies the (max( f L , f R ))-local biclique cover condition, but the converse is not always true.…”

Section: Model I: a Node Belongs To A Small Number Of Bicliquesmentioning

confidence: 99%

“…Finally, a problem related to our work here, is the local biclique cover problem [16], where for a given (not necessarily bipartite) graph G and a parameter f we want to determine whether there exists a biclique cover of the edges of G (using an arbitrary number of bicliques) in which each node belongs to at most f bicliques. Arora et al [17] studied a generalization of clique cover with applications to community detection where nodes are limited to participate in a few cliques.…”

Section: Introductionmentioning

confidence: 99%

Efficient Approximation for Restricted Biclique Cover Problems

Epasto

Upfal

2018

Algorithms

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Abstract:Covering the edges of a bipartite graph by a minimum set of bipartite complete graphs (bicliques) is a basic graph theoretic problem, with numerous applications. In particular, it is used to characterize parsimonious models of a set of observations (each biclique corresponds to a factor or feature that relates the observations in the two sets of nodes connected by the biclique). The decision version of the minimum biclique cover problem is NP-Complete, and unless P = NP, the cover size cannot be approximated in general within less than a sub-linear factor of the number of nodes (or edges) in the graph. In this work, we consider two natural restrictions to the problem, motivated by practical applications. In the first case, we restrict the number of bicliques a node can belong to. We show that when this number is at least 5, the problem is still NP-hard. In contrast, we show that when nodes belong to no more than two bicliques, the problem has efficient approximations. The second model we consider corresponds to observing a set of independent samples from an unknown model, governed by a possibly large number of factors. The model is defined by a bipartite graph G = (L, R, E), where each node in L is assigned to an arbitrary subset of up to a constant f factors, while the nodes in R (the independent observations) are assigned to random subsets of the set of k factors where k can grow with size of the graph. We show that this practical version of the biclique cover problem is amenable to efficient approximations.

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Local Clique Covering of Claw-Free Graphs

Cited by 12 publications

References 23 publications

Clique coverings and claw-free graphs

Clique coverings and claw-free graphs

Edge clique covers in graphs with independence number two

Efficient Approximation for Restricted Biclique Cover Problems

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