Bounding the Number of Minimal Dominating Sets: A Measure and Conquer Approach

Algorithms and Computation - Lecture Notes in Computer Science ◽

10.1007/11602613_58 ◽

2005 ◽

pp. 573-582 ◽

Cited By ~ 15

Author(s):

Fedor V. Fomin ◽

Fabrizio Grandoni ◽

Artem V. Pyatkin ◽

Alexey A. Stepanov

Keyword(s):

Dominating Sets ◽

Measure And Conquer

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Separate, Measure and Conquer: Faster Polynomial-Space Algorithms for Max 2-CSP and Counting Dominating Sets

Automata, Languages, and Programming - Lecture Notes in Computer Science ◽

10.1007/978-3-662-47672-7_46 ◽

2015 ◽

pp. 567-579 ◽

Cited By ~ 2

Author(s):

Serge Gaspers ◽

Gregory B. Sorkin

Keyword(s):

Polynomial Space ◽

Dominating Sets ◽

Measure And Conquer

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Domination cover number of graphs

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830919500204 ◽

2019 ◽

Vol 11 (02) ◽

pp. 1950020

Author(s):

M. Alambardar Meybodi ◽

M. R. Hooshmandasl ◽

P. Sharifani ◽

A. Shakiba

Keyword(s):

Real World ◽

Dominating Set ◽

Dominating Sets ◽

Art Gallery ◽

Art Gallery Problem ◽

Measure And Conquer ◽

Graph Classes ◽

Block Graphs ◽

New Criterion ◽

Real World Problems

A set [Formula: see text] for the graph [Formula: see text] is called a dominating set if any vertex [Formula: see text] has at least one neighbor in [Formula: see text]. Fomin et al. [Combinatorial bounds via measure and conquer: Bounding minimal dominating sets and applications, ACM Transactions on Algorithms (TALG) 5(1) (2008) 9] gave an algorithm for enumerating all minimal dominating sets with [Formula: see text] vertices in [Formula: see text] time. It is known that the number of minimal dominating sets for interval graphs and trees on [Formula: see text] vertices is at most [Formula: see text]. 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 [Formula: see text], denoted by [Formula: see text], is the summation of the degrees of the vertices in [Formula: see text]. Maximizing or minimizing this parameter among all minimal dominating sets has 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 and block graphs.

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