Secure Restrained Domination in Graphs

Pushpam, P. Roushini Leely; Suseendran, Chitra

doi:10.1007/s11786-015-0230-4

Cited by 11 publications

(3 citation statements)

References 8 publications

(7 reference statements)

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“…For weak Roman domination, results are known for various graphs based on chessboards [30], Cartesian products involving complete graphs [37], and Helm graphs and Web graphs [23]. For secure domination, results are known for some grid and torus graphs (for n ≤ 3) [17], various Cartesian products of stars, cycles, paths and complete graphs [37], and middle graphs [31].…”

Section: Introductionmentioning

confidence: 99%

Variants of the domination number for flower snarks

2024

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We consider the flower snarks, a widely studied infinite family of 3-regular graphs. For the Flower snark J n on 4n vertices, it is trivial to show that the domination number of J n is equal to n. However, results are more difficult to determine for variants of domination. The Roman domination, weakly convex domination, and convex domination numbers have been determined for flower snarks in previous works. We add to this literature by determining the independent domination, 2-domination, total domination, connected domination, upper domination, secure domination and weak Roman domination numbers for flower snarks.

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Section: Introductionmentioning

confidence: 99%

Variants of the domination number for flower snarks

2024

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“…A subset D of V (G) is a domination set if every vertex outside D has a neighbor in D. The domination number of G, denoted by γ(G), is the minimum cardinality of a domination set of G. A domination set D of G is a restrained domination set if every vertex outside D is adjacent to another vertex in V (G) \ D. The restrained domination number of G, denoted by γ r (G), is the minimum cardinality of a restrained domination set in G. The concept of the restrained domination was formally introduced in [5]. Since then, the variants of restrained domination have already been studied, the details refer to [3,14,15,17]. Double Roman domination is a stronger version of Roman domination, which was introduced in [2].…”

Section: Introductionmentioning

confidence: 99%

The restrained double Roman domination and graph operations

Gao¹,

Xi²,

Yue³

2021

Preprint

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Let G = (V (G), E(G)) be a simple graph. A restrained double Roman dominating function (RDRD-function) of G is a function f : V (G) → {0, 1, 2, 3} satisfying the following properties: if f (v) = 0, then the vertex v has at least two neighbours assigned 2 under f or one neighbour u with f (u) = 3; and if f (v) = 1, then the vertex v must have one neighbor u with f (u) ≥ 2; the induced graph by vertices assigned 0 under f contains no isolated vertex. The weight of a RDRD-function f is the sum f (V ) = v∈V (G) f (v), and the minimum weight of a RDRD-function on G is the restrained double Roman domination number (RDRD-number) of G, denoted by γ rdR (G). In this paper, we first prove that the problem of computing RDRD-number is NP-hard even for chordal graphs. And then we study the impact of some graph operations, such as strong product, cardinal product and corona with a graph, on restrained double Roman domination number.

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“…For weak Roman domination, results are known for various graphs based on chessboards [29], Cartesian products involving complete graphs [35], and Helm graphs and Web graphs [22]. For secure domination, results are known for some grid and torus graphs (for n ≤ 3) [15], various Cartesian products of stars, cycles, paths and complete graphs [35], and middle graphs [28].…”

Section: Introductionmentioning

confidence: 99%

Variants of the Domination Number for Flower Snarks

Burdett,

Haythorpe,

Newcombe

2021

Preprint

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We consider the flower snarks, a widely studied infinite family of 3-regular graphs. For the Flower snark Jn on 4n vertices, it is trivial to show that the domination number of Jn is equal to n. However, results are more difficult to determine for variants of domination. The Roman domination, weakly convex domination, and convex domination numbers have been determined for flower snarks in previous works. We add to this literature by determining the independent domination, 2-domination, total domination, connected domination, upper domination, secure Domination and weak Roman domination numbers for flower snarks.

show abstract

Secure Restrained Domination in Graphs

Cited by 11 publications

References 8 publications

Variants of the domination number for flower snarks

Variants of the domination number for flower snarks

The restrained double Roman domination and graph operations

Variants of the Domination Number for Flower Snarks

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