Graphs without a partition into two proportionally dense subgraphs

Bazgan, Cristina; Chlebíková, Janka; Dallard, Clément

doi:10.1016/j.ipl.2019.105877

Cited by 4 publications

(12 citation statements)

References 9 publications

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“…In our paper, we will use this version as well and call such a partition a generalised k-community structure. Notice that in [2], the authors introduce the notion of proportionally dense subgraph (PDS), and in particular they introduce so-called 2-PDS partitions, which correspond exactly to generalised 2-community structures. Our notion of generalised k-community structure can therefore be seen as a generalisation of 2-PDS partition to k-PDS partition.…”

Section: Introductionmentioning

confidence: 99%

“…Our notion of generalised k-community structure can therefore be seen as a generalisation of 2-PDS partition to k-PDS partition. The authors of [2] present two infinite families of graphs: (i) one infinite family of graphs that do not contain any generalised 2-community structure; (ii) one infinite family of graphs that do not admit any connected generalised 2-community structure (but a disconnected one).…”

Section: Introductionmentioning

confidence: 99%

“…Finally, we present an infinite family of connected graphs not admitting any generalized 2-community structure. In contrast to [2], where another such family has been presented but with the restriction that every graph of the family has an even number of vertices, our family contains graphs with an even number of vertices and graphs with an odd number of vertices (see Section 5).…”

Section: Introductionmentioning

confidence: 99%

“…Finally, we introduce a new infinite family of connected graphs that do not admit any 2-community structure (even if communities are allowed to have size one). Such a family was presented in Bazgan et al [2], but its graphs all contained an even number of vertices. The graphs in our new family may contain an even or an odd number of vertices.2010 Mathematics Subject Classification.…”

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confidence: 99%

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Finding $k$-community structures in special graph classes

Baghirova¹,

Dallard²,

Ries³

et al. 2022

Preprint

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For a fixed integer k ≥ 2, a k-community structure in an undirected graph is a partition of its vertex set into k sets (called communities), each of size at least two, such that every vertex of the graph has proportionally at least as many neighbours in its own community as in any other community. In this paper, we present a necessary and sufficient condition for a forest F to admit a k-community structure, for any integer k ≥ 2. Furthermore, if such a k-community exists, it can be found in polynomial time. This generalises a result of Bazgan et al. [3], who showed that all trees of size at least four, except stars, admit a 2-community that can be found in polynomial time. We also show that, if communities are allowed to have size one, then every forest with at least k ≥ 2 vertices admits a k-community structure that can be found in polynomial time. We then consider threshold graphs and show that every such connected graph admits a 2-community structure if and only if it is not isomorphic to a star; also, if such a 2-community structure exists, it can be found in polynomial time. Finally, we introduce a new infinite family of connected graphs that do not admit any 2-community structure (even if communities are allowed to have size one). Such a family was presented in Bazgan et al. [2], but its graphs all contained an even number of vertices. The graphs in our new family may contain an even or an odd number of vertices.2010 Mathematics Subject Classification. 05C85 .

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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Finding $k$-community structures in special graph classes

Baghirova¹,

Dallard²,

Ries³

et al. 2022

Preprint

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“…The special case where the community structure contains exactly two communities, namely a 2-community structure, has been studied in several classes of graphs: a 2-community structure always exists and can be found in polynomial time in trees, graphs with maximum degree 3, minimum degree |V | − 3, and complements of bipartite graphs [5]. Recently, the notion of 2-community structure has been studied under the name of 2-PDS partition [4]. In this paper, the authors described an infinite family of graphs without a 2-PDS partition, and a second infinite family of graphs without a connected 2-PDS partition (but with a disconnected one).…”

Section: Introductionmentioning

confidence: 99%

Proportionally dense subgraph of maximum size: Complexity and approximation

Bazgan

Chlebíková

Dallard

et al. 2019

Discrete Applied Mathematics

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We define a proportionally dense subgraph (PDS) as an induced subgraph of a graph with the property that each vertex in the PDS is adjacent to proportionally as many vertices in the subgraph as in the graph. We prove that the problem of finding a PDS of maximum size is APX-hard on split graphs, and NP-hard on bipartite graphs. We also show that deciding if a PDS is inclusion-wise maximal is co-NP-complete on bipartite graphs. Nevertheless, we present a simple polynomialtime (2− 2 ∆+1 )-approximation algorithm for the problem, where ∆ is the maximum degree of the graph. Finally, we show that all Hamiltonian cubic graphs with n vertices (except two) have a PDS of size ⌊ 2n+1 3 ⌋, which we prove to be an upper bound on the size of a PDS in cubic graphs.

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