On the number of minimal separators in graphs

Gaspers, Serge; Mackenzie, Simon

doi:10.1002/jgt.22179

Cited by 10 publications

(6 citation statements)

References 27 publications

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“…In general, a graph may have exponentially many minimal separators, and in fact there exist graphs with (3 n/3 ) minimal separators [12]. Recently, this bound was improved to (1.4457 n ) [14]. On the other hand, some graph classes have only polynomially (even linearly) many minimal separators.…”

Section: Related Workmentioning

confidence: 99%

On the maximum weight minimal separator

Hanaka

Bodlaender

Zanden

et al. 2019

Theoretical Computer Science

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Section: Related Workmentioning

confidence: 99%

On the maximum weight minimal separator

Hanaka

Bodlaender

Zanden

et al. 2019

Theoretical Computer Science

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“…We denote the set of minimal separators in G by S(G). Minimal separators are a commonly studied aspect of many graph classes [6,22,26,36]. Two particularly relevant cases include the fact that chordal graphs have at most n minimal separators [26], and that ciruclar-arc graphs have at most 2n 2 − 3n minimal separators [36].…”

Section: Minimal Separatorsmentioning

confidence: 99%

On H-Topological Intersection Graphs

Chaplick

Töpfer

Voborník

et al. 2017

Graph-Theoretic Concepts in Computer Science

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Biró, Hujter, and Tuza (1992) introduced the concept of H-graphs, intersection graphs of connected subregions of a graph H thought of as a one-dimensional topological space. They are related to and generalize many important classes of geometric intersection graphs, e.g., interval graphs, circular-arc graphs, split graphs, and chordal graphs. Our paper starts a new line of research in the area of geometric intersection graphs by studying H-graphs from the point of view of computational problems that are fundamental in theoretical computer science: recognition, graph isomorphism, dominating set, clique, and colorability.Surprisingly, we negatively answer the 25-year-old question of Biró, Hujter, and Tuza which asks whether H-graphs can be recognized in polynomial time, for a fixed graph H. We prove that it is NP-complete if H contains the diamond graph as a minor. On the positive side, we provide a polynomial-time algorithm recognizing T -graphs, for each fixed tree T . For the special case when T is a star S d of degree d, we have an O(n 3.5 )-time algorithm.We give FPTand XP-time algorithms solving the minimum dominating set problem on S dgraphs and H-graphs parametrized by d and the size of H, respectively. As a byproduct, the algorithm for H-graphs adapts to an XP-time algorithm for the independent set and the independent dominating set problems on H-graphs.If H contains the double-triangle as a minor, we prove that the graph isomorphism problem is GI-complete and that the clique problem is APX-hard. On the positive side, we show that the clique problem can be solved in polynomial time if H is a cactus graph. Also, when a graph G has a Helly H-representation, the clique problem can be solved in polynomial time.Further, we show that both the k-clique and the list k-coloring problems are solvable in FPTtime on H-graphs (parameterized by k and the treewidth of H). In fact, these results apply to classes graphs of graphs with treewidth bounded by a function of the clique number.Finally, we observe that H-graphs have at most n O( H ) minimal separators which allows us to apply the meta-algorithmic framework of Fomin, Todinca, and Villanger (2015) to show that for each fixed t, finding a maximum induced subgraph of treewidth t can be done in polynomial time. In the case when H is a cactus, we improve the bound to O( H n 2 ).

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“…Given this, it is important to know how many minimal separators graphs have. Gaspers and Mackenzie [19] showed that every graph with n vertices has O * (τ n ) minimal separators, where τ = 1.618 • • • is the golden ratio 2 . They showed that there is a graph with Ω(1.4422 n ) minimal separators.…”

Section: Introductionmentioning

confidence: 99%

Finding a Maximum Minimal Separator: Graph Classes and Fixed-Parameter Tractability

Hanaka¹,

Kobayashi²,

Kobayashi³

et al. 2020

Preprint

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We study the problem of finding a maximum cardinality minimal separator of a graph. This problem is known to be NP-hard even for bipartite graphs. In this paper, we strengthen this hardness by showing that for planar bipartite graphs, the problem remains NP-hard. Moreover, for co-bipartite graphs and for line graphs, the problem also remains NP-hard. On the positive side, we give an algorithm deciding whether an input graph has a minimal separator of size at least k that runs in time 2 O(k) n O(1) . We further show that a subexponential parameterized algorithm does not exist unless the Exponential Time Hypothesis (ETH) fails. Finally, we discuss a lower bound for polynomial kernelizations of this problem.

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On the number of minimal separators in graphs

Cited by 10 publications

References 27 publications

On the maximum weight minimal separator

On the maximum weight minimal separator

On H-Topological Intersection Graphs

Finding a Maximum Minimal Separator: Graph Classes and Fixed-Parameter Tractability

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