Approximation Scheme for Lowest Outdegree Orientation and Graph Density Measures

Kowalik, Łukasz

doi:10.1007/11940128_56

Cited by 31 publications

(34 citation statements)

References 19 publications

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“…We could prove Lemma 5 directly with a simple counting argument on the indegrees of nodes in R k (see Appendix A) or by using a network flow construction similar to Goldberg's and the max flow-min cut theorem [18]. The upper bound given in Lemma 6 may be proven directly by using a counting argument for the indegrees of vertices in an egalitarian orientation of the densest subnetwork (see Appendix A) or by using the relationship between the density of the maximum density subgraph and the psuedoarboricity [24].…”

Section: Density and The Density Decompositionmentioning

confidence: 99%

Density Decompositions of Networks

Borradaile

Migler

Wilfong

2018

Complex Networks IX

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We introduce a new topological descriptor of a network called the density decomposition which is a partition of the nodes of a network into regions of uniform density. The decomposition we define is unique in the sense that a given network has exactly one density decomposition. The number of nodes in each partition defines a density distribution which we find is measurably similar to the degree distribution of given real networks (social, internet, etc.) and measurably dissimilar in synthetic networks (preferential attachment, small world, etc.).

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Section: Density and The Density Decompositionmentioning

confidence: 99%

Density Decompositions of Networks

Borradaile

Migler

Wilfong

2018

Complex Networks IX

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“…[11] for some relations between classes of sparse graphs). We show that any n-vertex twinless graph of arboricity c contains an induced matching of size Ω( 1 c n 1/c ).…”

Section: E(g[j])| |J|mentioning

confidence: 99%

Improved induced matchings in sparse graphs

Erman¹,

Kowalik²,

Krnc³

et al. 2010

Discrete Applied Mathematics

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a b s t r a c tAn induced matching in a graph G = (V , E) is a matching M such that (V , M) is an induced subgraph of G. Clearly, among two vertices with the same neighbourhood (called twins) at most one is matched in any induced matching, and if one of them is matched then there is another matching of the same size that matches the other vertex. Motivated by this, Kanj et al.[10] studied induced matchings in twinless graphs. They showed that any twinless planar graph contains an induced matching of size at least n 40 and that there are twinless planar graphs that do not contain an induced matching of size greater than n 27 + O(1). We improve both these bounds to n 28 + O(1), which is tight up to an additive constant. This implies that the problem of deciding whether a planar graph has an induced matching of size k has a kernel of size at most 28k. We also show for the first time that this problem is fixed parameter tractable for graphs of bounded arboricity. Kanj et al. also presented an algorithm which decides in O(2 159 √ k + n)-time whether an n-vertex planar graph contains an induced matching of size k. Our results improve the time complexity analysis of their algorithm. However, we also show a more efficient O(2 25.5 √ k + n)-time algorithm. Its main ingredient is a new, O * (4 l )-time algorithm for finding a maximum induced matching in a graph of branch width at most l.

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“…Edge Orientation Problem. Since Venkateswaran [27] introduced the problem, (centralized) polynomial-time algorithms are known to give optimal solutions for unweighted graphs [3], [21], [27]. However, a series of works [2], [3] showed that the weighted version is NP-hard even when all edge weights belong to the set {1, k}, where k is any fixed integer greater than 1; on the other hand, for integer edge weights, (2 − 1 k )-approximation can be achieved, where k is the maximum weight.…”

Section: Min-maxmentioning

confidence: 99%