Connected dominating sets and connected domination polynomials of square of centipedes

Vijayan, A.; Mabel, M.Felix Nes

doi:10.1080/02522667.2015.1103031

Cited by 7 publications

(3 citation statements)

References 2 publications

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“…For total domination, results are known for some grid graphs G(n, m) (for n ≤ 6) [15,21], Knödel graphs [27], and various graphs from chemistry [26]. For connected domination, results are known for trees [33], circulant graphs [35], Centipede graphs [38], and various graphs from chemistry [26]. 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].…”

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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“…For total domination, results are known for some grid graphs G(n, m) (for n ≤ 6) [13,20], Knödel graphs [19], and various graphs from chemistry [25]. For connected domination, results are known for trees [31], circulant graphs [33], Centipede graphs [36], and various graphs from chemistry [25]. 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].…”

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.

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“…Domination polynomials, independent polynomials, and independent dominating polynomials have been of recent interest; the interested reader should see [6][7][8][9], and [10]. For example Alikhani observed that evaluating the domination (independent or independent dominating) polynomial at 1 will yield the number of dominating (independent or independent dominating) sets of a graph.…”

Section: Introductionmentioning

confidence: 99%

Graph polynomials for a class of DI-pathological graphs

Hammer

Harrington

2020

AKCE International Journal of Graphs and Combinatorics

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Let G = (V, E) be a simple graph. A dominating set D ⊆ V is a set such that the closed neighborhood of D is the entire vertex set. An independence set of a graph G is a subset of vertices that are pairwise non-adjacent. A DI-pathological graph is a graph where every minimum dominating set intersects every maximal independent set. Let d(G, i) denote the number of dominating sets of G of size i. The domination polynomial of a graph G is defined by D(G, x) = ∑ |V | i=γ (G) d(G, i)x i. Let s(G, i) denote the number of independent sets of size i in a graph G. The independence polynomial is defined by I (G, x) = ∑ α(G) i=0 s(G, i)x i. In this paper, we will examine the domination polynomial and the independence polynomial of a family of extremal DI-pathological graphs. We will further define an independent dominating set and examine the corresponding independent domination polynomial for these graphs.

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Connected dominating sets and connected domination polynomials of square of centipedes

Cited by 7 publications

References 2 publications

Variants of the domination number for flower snarks

Variants of the domination number for flower snarks

Variants of the Domination Number for Flower Snarks

Graph polynomials for a class of DI-pathological graphs

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