Tight bounds for eternal dominating sets in graphs

Goldwasser, John L.; Klostermeyer, William F.

doi:10.1016/j.disc.2007.06.005

Cited by 33 publications

(15 citation statements)

References 4 publications

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“…This example can be generalized as follows to obtain a class of graphs G such that γ ∞ m (G) = 2 and α(G) = 3. In C 6 = v 0 , v 1 , ..., v 5 , v 0 , replace each v i by a complete graph H i of any order, and join each vertex of H i , i = 0, ..., 5, to each vertex of H i+1 (mod 6) and to each vertex of H i−1 (mod 6) to form the graph H. Note that α(H) = 3 and, by…”

Section: M-eternal Domination and Independencementioning

confidence: 99%

Domination, eternal domination and clique covering

Klostermeyer¹,

Mynhardt²

2015

Discuss. Math. Graph Theory

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Eternal and m-eternal domination are concerned with using mobile guards to protect a graph against infinite sequences of attacks at vertices. Eternal domination allows one guard to move per attack, whereas more than one guard may move per attack in the m-eternal domination model. Inequality chains consisting of the domination, eternal domination, m-eternal domination, independence, and clique covering numbers of graph are explored in this paper.Among other results, we characterize bipartite and triangle-free graphs with domination and eternal domination numbers equal to two, trees with equal m-eternal domination and clique covering numbers, and two classes of graphs with equal domination, eternal domination and clique covering numbers.

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Section: M-eternal Domination and Independencementioning

confidence: 99%

Domination, eternal domination and clique covering

Klostermeyer¹,

Mynhardt²

2015

Discuss. Math. Graph Theory

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“…This result has recently been improved to O * (1.5673 n ) [22], which can be lowered to O * (1.5014 n ) at the cost of exponential space [22]. Moreover, the Roman Domination problem can be related to several other variants of defense-like domination, such as secure domination (see, e.g., [3,4,11]), or eternal domination (see, e.g., [9,10]). …”

Section: Introductionmentioning

confidence: 99%

Exact Algorithms for Weak Roman Domination

Chapelle¹,

Cochefert

Couturier

et al. 2013

Lecture Notes in Computer Science

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International audienceWe consider the Weak Roman Domination problem. Given an undirected graph G = (V,E), the aim is to find a weak roman domination function (wrd-function for short) of minimum cost, i.e. a function f: V → {0,1,2} such that every vertex v ∈ V is defended (i.e. there exists a neighbor u of v, possibly u = v, such that f(u)≥1) and for every vertex v ∈ V with f(v) = 0 there exists a neighbor u of v such that f(u)≥1 and the function f_u → v defined by:f_u→v(x)={1 if x=v,f(u)-1 if x=u,f(x) if x∉{u,v}} does not contain any undefended vertex. The cost of a wrd-function f is defined by cost(f) = ∑ _v ∈ V f(v). The trivial enumeration algorithm runs in time O^∗(3ⁿ) and polynomial space and is the best one known for the problem so far. We are breaking the trivial enumeration barrier by providing two faster algorithms: we first prove that the problem can be solved in O^∗(2ⁿ) time needing exponential space, and then describe an O^∗(2.2279ⁿ) algorithm using polynomial space. Our results rely on structural properties of a wrd-function, as well as on the best polynomial space algorithm for the Red-Blue Dominating Set problem

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“…The problem of finding the number of guards required for the eternal security of graphs has been previously studied. Goddard et al [2] related this number to the domination number of a graph, whereas [3,6] provided bounds in terms of the independence number of a graph. In [7], the required number of guards is compared to the vertex cover number.…”

Section: Introductionmentioning

confidence: 99%

Guarding networks through heterogeneous mobile guards

Abbas

Bhatia

Koutsoukos

2015

2015 American Control Conference (ACC)

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Abstract-In this article, the issue of guarding multi-agent systems against a sequence of intruder attacks through mobile heterogeneous guards (guards with different ranges) is discussed. The article makes use of graph theoretic abstractions of such systems in which agents are the nodes of a graph and edges represent interconnections between agents. Guards represent specialized mobile agents on specific nodes with capabilities to successfully detect and respond to an attack within their guarding range. Using this abstraction, the article addresses the problem in the context of eternal security problem in graphs. Eternal security refers to securing all the nodes in a graph against an infinite sequence of intruder attacks by a certain minimum number of guards. This paper makes use of heterogeneous guards and addresses all the components of the eternal security problem including the number of guards, their deployment and movement strategies. In the proposed solution, a graph is decomposed into clusters and a guard with appropriate range is then assigned to each cluster. These guards ensure that all nodes within their corresponding cluster are being protected at all times, thereby achieving the eternal security in the graph.

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Tight bounds for eternal dominating sets in graphs

Cited by 33 publications

References 4 publications

Domination, eternal domination and clique covering

Domination, eternal domination and clique covering

Exact Algorithms for Weak Roman Domination

Guarding networks through heterogeneous mobile guards

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