A Dichotomy for Upper Domination in Monogenic Classes

AbouEisha, Hassan; Hussain, Shahid; Monnot, Jérôme; Ries, Bernard

doi:10.1007/978-3-319-12691-3_20

Cited by 8 publications

(33 citation statements)

References 14 publications

(15 reference statements)

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“…We then obtain the inapproximability of Upper Domination by performing a reduction from an instance with sufficiently large d. We also show that Upper Domination remains hard on two restricted cases: the problem is still APX-hard on cubic graphs, and NP-hard on planar subcubic graphs. Since the problem is easy on graphs of maximum degree 2, our results completely characterise the complexity of the problem in terms of maximum degree (the best previously known result was NP-hardness for planar graphs of maximum degree 6 [1]).…”

Section: Introductionmentioning

confidence: 67%

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Upper Domination: Complexity and Approximation

Bazgan¹,

Branković²,

Casel³

et al. 2016

Lecture Notes in Computer Science

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International audienceWe consider Upper Domination, the problem of finding a maximum cardinality minimal dominating set in a graph. We show that this problem does not admit an $n^{1-\epsilon}$ approximation for any $\epsilon>0$, making it significantly harder than Dominating Set, while it remains hard even on severely restricted special cases, such as cubic graphs (APX-hard), and planar subcubic graphs (NP-hard). We complement our negative results by showing that the problem admits an $O(\Delta)$ approximation on graphs of maximum degree $\Delta$, as well as an EPTAS on planar graphs. Along the way, we also derive essentially tight $n^{1-1/d} upper and lower bounds on the approximability of the related problem Maximum Minimal Hitting Set on d-uniform hypergraphs, generalising known results for Maximum Minimal Vertex Cover

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Section: Introductionmentioning

confidence: 67%

“…It has long been known that Upper Domination is NP-complete in general [10], and even for graphs of maximum degree 6 [1]. Some polynomial-time solvable graph classes are also known.…”

Section: Previous Resultsmentioning

confidence: 99%

Upper Domination: Complexity and Approximation

Bazgan¹,

Branković²,

Casel³

et al. 2016

Lecture Notes in Computer Science

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“…In this paper, we identified the first boundary class for the upper dominating set problem. Since the problem is NP-hard in the class of triangle-free graphs [1], we known (by Theorem 1) that there must exist at least one more boundary class for the problem. Revealing this class is a challenging open question.…”

Section: Resultsmentioning

confidence: 99%

“…We complete this part of the section with the following technical lemma, proved in [1], where a private neighbour of a vertex x ∈ D is a vertex y ∈ D such that x is the only neighbour of y in D.…”

Section: Preliminariesmentioning

confidence: 99%

“…the problem of finding an upper dominating set in a graph) is known to be NP-hard [5]. Moreover, it remains difficult under substantial restrictions, for instance, for triangle-free graphs and the complements of bipartite graphs [1]. On the other hand, in some particular graph classes, the problem can be solved in polynomial time, which is the case for bipartite graphs [6], chordal graphs [10], generalized series-parallel graphs [9], graphs of bounded clique-width [7] and 2K 2 -free graphs [1].…”

Section: Introductionmentioning

confidence: 99%

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A Boundary Property for Upper Domination

AbouEisha

Hussain

Monnot

et al. 2016

Lecture Notes in Computer Science

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LNCS n°9843International audienceAn upper dominating set in a graph is a minimal (with respect to set inclusion) dominating set of maximum cardinality. The problem of finding an upper dominating set is generally NP-hard, but can be solved in polynomial time in some restricted graph classes, such as P4-free graphs or 2K2-free graphs. For classes defined by finitely many forbidden induced subgraphs, the boundary separating difficult instances of the problem from polynomially solvable ones consists of the so called boundary classes. However, none of such classes has been identified so far for the upper dominating set problem. In the present paper, we discover the first boundary class for this problem

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Upper Domination: Towards a Dichotomy Through Boundary Properties

et al. 2017

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An upper dominating set in a graph is a minimal (with respect to set inclusion) dominating set of maximum cardinality. The problem of finding an upper dominating set is generally NP-hard. We study the complexity of this problem in classes of graphs defined by finitely many forbidden induced subgraphs and conjecture that the problem admits a dichotomy in this family, i.e. it is either NP-hard or polynomial-time solvable for each class in the family. A helpful tool to study the complexity of an algorithmic problem on finitely defined classes of graphs is the notion of boundary classes. However, none of such classes has been identified so far for the upper dominating set problem. In the present paper, we discover the first boundary class for this problem and prove the dichotomy for classes defined by a single forbidden induced subgraph.

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A Dichotomy for Upper Domination in Monogenic Classes

Cited by 8 publications

References 14 publications

Upper Domination: Complexity and Approximation

Upper Domination: Complexity and Approximation

A Boundary Property for Upper Domination

Upper Domination: Towards a Dichotomy Through Boundary Properties

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