On the number of minimal dominating sets on some graph classes

Couturier, Jean-François; Letourneur, Romain; Liedloff, Mathieu

doi:10.1016/j.tcs.2014.11.006

Cited by 31 publications

(20 citation statements)

References 6 publications

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“…Partial Dominating Set can be solved in time O(1.5673 n ) using polynomial space [52] and in time O(1.5012 n ) using exponential space [52]. Other variants have been considered (see, e.g., [45,61]) and the problems have been studied on many graph classes (see, e.g., [10,11,28,34,43,46,53]). The #DS problem is to compute, for a given graph G, the function d : {0, .…”

Section: Counting Dominating Setsmentioning

confidence: 99%

Separate, Measure and Conquer

Gaspers

Sorkin

2017

ACM Trans. Algorithms

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We show a method resulting in the improvement of several polynomial-space, exponential-time algorithms. The method capitalizes on the existence of small balanced separators for sparse graphs, which can be exploited for branching to disconnect an instance into independent components. For this algorithm design paradigm, the challenge to date has been to obtain improvements in worst-case analyses of algorithms, compared with algorithms that are analyzed with advanced methods, notably Measure and Conquer. Our contribution is the design of a general method to integrate the advantage from the separator-branching into Measure and Conquer, for a more precise and improved running time analysis. We illustrate the method with improved algorithms for Max (r , 2)-CSP and #Dominating Set. An instance of the problem Max (r , 2)-CSP, or simply Max 2-CSP, is parameterized by the domain size r (often 2), the number of variables n (vertices in the constraint graph G), and the number of constraints m (edges in G). When G is cubic, and omitting sub-exponential terms here for clarity, we give an algorithm running in time r (1/5) n = r (2/15) m ; the previous best was r (1/4) n = r (1/6) m. By known results, this improvement for the cubic case results in an algorithm running in time r (9/50) m for general instances; the previous best was r (19/100) m. We show that the analysis of the earlier algorithm was tight: our improvement is in the algorithm, not just the analysis. The same running time improvements hold for Max Cut, an important special case of Max 2-CSP, and for Polynomial and Ring CSP, generalizations encompassing graph bisection, the Ising model, and counting. We also give faster algorithms for #Dominating Set, counting the dominating sets of every cardinality 0,. .. , n for a graph G of order n. For cubic graphs, our algorithm runs in time 3 (1/5) n ; the previous best was 2 (1/2) n. For general graphs, we give an unrelated algorithm running in time 1.5183 n ; the previous best was 1.5673 n. The previous best algorithms for these problems all used local transformations and were analyzed by the Measure and Conquer method. Our new algorithms capitalize on the existence of small balanced separators for cubic graphs-a non-local property-and the ability to tailor the local algorithms always to "pivot" on a vertex in the separator. The new algorithms perform much as the old ones until the separator is empty, at which point they gain because the remaining vertices are split into two independent problem instances that can be solved recursively. It is likely that such algorithms can be effective for other problems too, and we present their design and analysis in a general framework. CCS Concepts: • Theory of computation → Graph algorithms analysis; Parameterized complexity and exact algorithms; Backtracking;

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Section: Counting Dominating Setsmentioning

confidence: 99%

Separate, Measure and Conquer

Gaspers

Sorkin

2017

ACM Trans. Algorithms

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“…They established a lower and an upper bound for the maximum number of minimal dominating sets in graphs. This gap has been narrowed on some well-known graph classes, such as chordal graphs [7], trees [14] and co-bipartite graphs [8]. Tight bounds have been obtained for some graph classes such as cographs and split graphs [7,8].…”

Section: Introductionmentioning

confidence: 99%

Domination cover number of graphs

Meybodi

Hooshmandasl

Sharifani

et al. 2019

Discrete Math. Algorithm. Appl.

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A set D ⊆ V for the graph G = (V, E) is called a dominating set if any vertex v ∈ V \ D has at least one neighbor in D. Fomin et al. [9] gave an algorithm for enumerating all minimal dominating sets with n vertices in O(1.7159 n ) time. It is known that the number of minimal dominating sets for interval graphs and trees on n vertices is at most 3 n/3 ≈ 1.4422 n . In this paper, we introduce the domination cover number as a new criterion for evaluating the dominating sets in graphs. The domination cover number of a dominating set D, denoted by CD(G), is the summation of the degrees of the vertices in D. Maximizing or minimizing this parameter among all minimal dominating sets have interesting applications in many real-world problems, such as the art gallery problem. Moreover, we investigate this concept for different graph classes and propose some algorithms for finding the domination cover number in trees, block graphs.

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“…Despite the vast applications, solving the MDS problem of a graph is a challenge, because finding a MDS of a graph is NP-hard [1]. Over the past years, the MDS problem has attracted considerable attention from theoretical computer science [10,11,12,13,14], discrete and combinatorial mathematics [15,16,17], as well as statistical physics [18], and continues to be an active object of research [19,20]. Extensive empirical research [21] uncovered that most real networks exhibit the prominent scale-free behavior [22], with the degree of their vertices following a power-law distribution P (k) ∼ k −γ .…”

Section: Introductionmentioning

confidence: 99%

Domination number and minimum dominating sets in pseudofractal scale-free web and Sierpiński graph

Shan

Zhang

2017

Theoretical Computer Science

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The minimum dominating set (MDS) problem is a fundamental subject of theoretical computer science, and has found vast applications in different areas, including sensor networks, protein interaction networks, and structural controllability. However, the determination of the size of a MDS and the number of all MDSs in a general network is NP-hard, and it thus makes sense to seek particular graphs for which the MDS problem can be solved analytically. In this paper, we study the MDS problem in the pseudofractal scale-free web and the Sierpiński graph, which have the same number of vertices and edges. For both networks, we determine explicitly the domination number, as well as the number of distinct MDSs. We show that the pseudofractal scale-free web has a unique MDS, and its domination number is only half of that for the Sierpiński graph, which has many MDSs. We argue that the scale-free topology is responsible for the difference of the size and number of MDSs between the two studied graphs, which in turn indicates that power-law degree distribution plays an important role in the MDS problem and its applications in scale-free networks.

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On the number of minimal dominating sets on some graph classes

Cited by 31 publications

References 6 publications

Separate, Measure and Conquer

Separate, Measure and Conquer

Domination cover number of graphs

Domination number and minimum dominating sets in pseudofractal scale-free web and Sierpiński graph

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