Lower Bounds on the Minus Domination and k-Subdomination Numbers

Kang, Lei; Qiao, Hong; Du, Ding‐Zhu

doi:10.1007/3-540-44679-6_41

Cited by 5 publications

(8 citation statements)

References 7 publications

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“…which coincides with (26) and is equivalent to the global support inequality, can be derived from a result of [16,17] (cf. [20,Theorem 4.15]), which in turn follows from the results in [19,21] and Theorem 2 in [22] on k-subdomination numbers, by substituting k = n+1 2 (in the notation of [17], q = n+1 2n ) in the case of odd n. As shown in Section 3, this bound is sharp for low-degree graphs. For high-degree graphs it is not sharp, and its inaccuracy in bounding h min (n, d) can be estimated by 1 − n+1 2d d+1 2 , which reduces to n 2 −1 4n when d = n. This follows from considering the difference between the bounds (3) and ( 1) and the propositions on the sharpness of bound (3) proved in Section 3.…”

Section: Alternative Formulation: Majority Domination On Regular Graphsmentioning

confidence: 74%

“…which is stronger in some cases than (25), although it is claimed [14,[16][17][18][19] that the bound (25) is sharp or best possible. Technically, it is sharp in a weak sense, that is, it cannot be improved for some values of n and d. Proposition 21 below states that for any odd d > 1 there are infinitely many n such that bound (25) can be improved for the class G n|d of d-regular graphs with loops on n vertices.…”

Section: Alternative Formulation: Majority Domination On Regular Graphsmentioning

confidence: 99%

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The Power of Small Coalitions under Two-Tier Majority on Regular Graphs

Chebotarev¹,

Peleg²

2022

Preprint

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In this paper, we study the following problem. Consider a setting where a proposal is offered to the vertices of a given network G, and the vertices must conduct a vote and decide whether to accept the proposal or reject it. Each vertex v has its own valuation of the proposal; we say that v is "happy" if its valuation is positive (i.e., it expects to gain from adopting the proposal) and "sad" if its valuation is negative. However, vertices do not base their vote merely on their own valuation. Rather, a vertex v is a proponent of the proposal if a majority of its neighbors are happy with it and an opponent in the opposite case. At the end of the vote, the network collectively accepts the proposal whenever a majority of its vertices are proponents. We study this problem on regular graphs with loops. Specifically, we consider the class G n|d|h of d-regular graphs of odd order n with all n loops and h happy vertices. We are interested in establishing necessary and sufficient conditions for the class G n|d|h to contain a labeled graph accepting the proposal, as well as conditions to contain a graph rejecting the proposal. We also discuss connections to the existing literature, including that on majority domination, and investigate the properties of the obtained conditions.

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Section: Alternative Formulation: Majority Domination On Regular Graphsmentioning

confidence: 74%

Section: Alternative Formulation: Majority Domination On Regular Graphsmentioning

confidence: 99%

The Power of Small Coalitions under Two-Tier Majority on Regular Graphs

Chebotarev¹,

Peleg²

2022

Preprint

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“…We do not know whether the minus total domination number of a triangle-free graph has the same lower bound as described in Theorem 7. Moreover, Kang et al [7] and Wang et al [9] independently gave sharp lower bound on the minus domination number for bipartite graphs. Kang et al [6] further extented the result to k-partite graphs.…”

Section: Resultsmentioning

confidence: 99%

Remarks on the minus (signed) total domination in graphs

Shan

Cheng

2008

Discrete Mathematics

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A function f : V (G) → {+1, 0, −1} defined on the vertices of a graph G is a minus total dominating function if the sum of its function values over any open neighborhood is at least one. The minus total domination number γ − t (G) of G is the minimum weight of a minus total dominating function on G. By simply changing "{+1, 0 − 1}" in the above definition to "{+1, −1}", we can define the signed total dominating function and the signed total domination number γ s t (G) of G. In this paper we present a sharp lower bound on the signed total domination number for a k-partite graph, which results in a short proof of a result due to Kang et al. on the minus total domination number for a k-partite graph. We also give sharp lower bounds on γ s t and γ − t for triangle-free graphs and characterize the extremal graphs achieving these bounds.

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“…It was introduced by J.E. Dunbar et al in [2], and has been studied by other authors in [3][4][5]. Quite analogously, the neighborhood of edge e is a set {e |e ∈ E(G) and e is adjacent to e}, which is denoted by N(e), and N[e] = N(e) ∪ {e} is the closed edge neighborhood of e. We call a function f : E(G) → {−1, 0, 1} a minus edge domination function (shortly MEDF) of G, if f [e] = f (N[e]) = e ∈N [e] f (e ) 1 for each e ∈ E(G).…”

Section: U∈nmentioning

confidence: 99%

On the Mixed Minus Domination in Graphs

Xiang-yang

2013

J. Oper. Res. Soc. China

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Lower Bounds on the Minus Domination and k-Subdomination Numbers

Cited by 5 publications

References 7 publications

The Power of Small Coalitions under Two-Tier Majority on Regular Graphs

The Power of Small Coalitions under Two-Tier Majority on Regular Graphs

Remarks on the minus (signed) total domination in graphs

On the Mixed Minus Domination in Graphs

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