(Total) Vector domination for graphs with bounded branchwidth

Ishii, Toshimasa; Ono, Hiroyuki; Uno, Yushi

doi:10.1016/j.dam.2016.03.002

Cited by 5 publications

(11 citation statements)

References 20 publications

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“…It follows that the vector domination for planar graphs is FPT with respect to solution size k * . For the total vector domination and multiple vector domination, [27] presents first polynomial time algorithms for graphs of bounded treewidth (or bounded branchwidth). The same paper presents first subexponential fixedparameter algorithms with respect to k * for apex-minor-free graphs for all the domination problems (i.e., vector, total vector, and multiple domination problems).…”

Section: Related Workmentioning

confidence: 99%

“…The same paper presents first subexponential fixedparameter algorithms with respect to k * for apex-minor-free graphs for all the domination problems (i.e., vector, total vector, and multiple domination problems). See also [26].…”

Section: Related Workmentioning

confidence: 99%

“…Here, we introduce the following basic properties about branch decompositions, which will be utilized in the subsequent sections (see e.g., [26,27,Lemma 2] for its proof).…”

Section: Lemma 1 ([13]) Let G Be An H-minor-free Graph Then a Branmentioning

confidence: 99%

“…Since cw(G) ≤ 3•2 tw(G)−1 holds where tw(G) and cw(G) respectively denote the treewidth and clique-width of graph G [9], their polynomial-time algorithm implies the polynomial-time solvability of the vector domination problem for graphs of bounded treewidth. Also, polynomial-time algorithms for the domination problems of graphs with bounded branchwidth are independently proposed in [26,27]. It is known that max{bw(G), 2} ≤ tw(G)+1 ≤ max{3bw(G)/2, 2}, where bw(G) denotes the branchwidth of graph G [30].…”

Section: Introductionmentioning

confidence: 99%

“…(2) There is an algorithm whose running time is O(3 3bw(G)/2 |E|). This idea can be extended to the domination problems, and there are subexponential fixed-parameter algorithms with respect to solution size for the domination problems of apexminor-free graphs, as mentioned above [26,27].…”

Section: Introductionmentioning

confidence: 99%

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Subexponential fixed-parameter algorithms for partial vector domination

Ishii

Ono

Uno

2016

Discrete Optimization

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Given a graph G = (V, E) of order n and an n-dimensional non-negative vector d = (d(1), d(2),. .. , d(n)), called demand vector, the vector domination (resp., total vector domination) is the problem of finding a minimum S ⊆ V such that every vertex v in V \ S (resp., in V) has at least d(v) neighbors in S. The (total) vector domination is a generalization of many dominating set type problems, e.g., the (total) dominating set problem, the (total) k-dominating set problem (this k is different from the solution size), and so on, and subexponential fixed-parameter algorithms with respect to solution size for apex-minor-free graphs (so for planar graphs) are known. In this paper, we consider maximization versions of the problems; that is, for a given integer k, the goal is to find an S ⊆ V with size k that maximizes the total sum of satisfied demands. For these problems, we design subexponential fixed-parameter algorithms with respect to k for apex-minor-free

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Section: Related Workmentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

“…Here, we introduce the following basic properties about branch decompositions, which will be utilized in the subsequent sections (see e.g., [26,27,Lemma 2] for its proof).…”

Section: Lemma 1 ([13]) Let G Be An H-minor-free Graph Then a Branmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Subexponential fixed-parameter algorithms for partial vector domination

Ishii

Ono

Uno

2016

Discrete Optimization

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Improved Algorithms for k-Domination and Total k-Domination in Proper Interval Graphs

Chiarelli

Hartinger

Leoni

et al. 2018

Lecture Notes in Computer Science

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Given a positive integer k, a k-dominating set in a graph G is a set of vertices such that every vertex not in the set has at least k neighbors in the set. A total k-dominating set, also known as a k-tuple total dominating set, is a set of vertices such that every vertex of the graph has at least k neighbors in the set. The problems of finding the minimum size of a k-dominating, resp. total k-dominating set, in a given graph, are referred to as k-domination, resp. total k-domination. These generalizations of the classical domination and total domination problems are known to be NP-hard in the class of chordal graphs, and, more specifically, even in the classes of split graphs (both problems) and undirected path graphs (in the case of total k-domination). On the other hand, it follows from recent work by Kang et al. (2017) that these two families of problems are solvable in time O(|V (G)| 6k+4 ) in the class of interval graphs. In this work, we develop faster algorithms for k-domination and total k-domination in the class of proper interval graphs. The algorithms run in time O(|V (G)| 3k ) for each fixed k ≥ 1 and are also applicable to the weighted case.

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Domination and convexity problems in the target set selection model

Araújo

Sampaio

2023

Discrete Applied Mathematics

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(Total) Vector domination for graphs with bounded branchwidth

Cited by 5 publications

References 20 publications

Subexponential fixed-parameter algorithms for partial vector domination

Subexponential fixed-parameter algorithms for partial vector domination

Improved Algorithms for k-Domination and Total k-Domination in Proper Interval Graphs

Domination and convexity problems in the target set selection model

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