Independent Domination in Cubic Graphs

Dorbec, Paul; Henning, Michael A.; Montassier, Mickael; Southey, Justin

doi:10.1002/jgt.21855

Cited by 20 publications

(45 citation statements)

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“…still open, and as a partial result, Dorbec et al [6] showed that the conjecture holds if in addition G does not have a subgraph isomorphic to K 2,3 . For other conjectures and partial results regarding the independent domination number of subclasses of cubic graphs, see [2,7,9,10].…”

Section: Introductionmentioning

confidence: 99%

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On independent domination of regular graphs

Cho¹,

Choi

Park

2021

Preprint

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Given a graph G, a dominating set of G is a set S of vertices such that each vertex not in S has a neighbor in S. The domination number of G, denoted γ(G), is the minimum size of a dominating set of G. The independent domination number of G, denoted i(G), is the minimum size of a dominating set of G that is also independent. Note that every graph has an independent dominating set, as a maximal independent set is equivalent to an independent dominating set.Let G be a connected k-regular graph that is not K k,k where k ≥ 4. Generalizing a result by Lam, Shiu, and Sun, we prove that i(G) ≤ k−1 2k−1 |V (G)|, which is tight for k = 4. This answers a question by Goddard et al. in the affirmative. We also show that i(G) γ(G) ≤ k 3 −3k 2 +2 2k 2 −6k+2 , strengthening upon a result of Knor, Škrekovski, and Tepeh. In addition, we prove that a graph G with maximum degree at most 4 satisfies i(G ) ≤ 5 9 |V (G )|, which is also tight.

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Section: Introductionmentioning

confidence: 99%

“…Using Theorem 1.3 and an idea in [20], we prove Theorem 1.4 in Section 3. In Section 4, we prove a statement slightly stronger than Theorem 1.5, where the proof adopts the approach of [6].…”

Section: Introductionmentioning

confidence: 99%

On independent domination of regular graphs

Cho¹,

Choi

Park

2021

Preprint

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“…We remark that several papers, see for example [1,2,3,11,19], in which upper bounds are obtained on the independent domination number of cubic graphs present more general results on subcubic graphs.…”

Section: Motivation and Known Resultsmentioning

confidence: 82%

“…Dorbec, Henning, Montassier, and Southey [3] proved Conjecture 1 in the case when there is no subgraph isomorphic to K 2,3 . In general, however, Conjecture 1 remains unresolved.…”

Section: Motivation and Known Resultsmentioning

confidence: 99%

“…A set S of vertices in a graph G is a dominating set if every vertex not in S is adjacent to a vertex in S. If, in addition, S is an independent set, then S is an independent dominating set, abbreviated ID-set. The independent domination number, denoted i(G), of G is the minimum cardinality of an ID-set in G. The concept of independent domination number of graphs is studied extensively in the literature, for example see [1,2,3,4,5,7,8,9,10,11,13,14,15,16,17,18,19,20]. A survey on independent domination in graphs can be found in [6].…”

Section: Introductionmentioning

confidence: 99%

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Independent Domination in Subcubic Graphs

Akbari

Doosthosseini

et al. 2020

Preprint

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A set S of vertices in a graph G is a dominating set if every vertex not in S is adjacent to a vertex in S. If, in addition, S is an independent set, then S is an independent dominating set. The independent domination number i(G) of G is the minimum cardinality of an independent dominating set in G. In 2013 Goddard and Henning [Discrete Math 313 (2013), 839-854] conjectured that if G is a connected cubic graph of order n, then i(G) ≤ 3 8 n, except if G is the complete bipartite graph K 3,3 or the 5-prism C 5 2 K 2 . Further they construct two infinite families of connected cubic graphs with independent domination three-eighths their order. They remark that perhaps it is even true that for n > 10 these two families are only families for which equality holds. In this paper, we provide a new family of connected cubic graphs G of order n such that i(G) = 3 8 n. We also show that if G is a subcubic graph of order n with no isolated vertex, then i(G) ≤ 1 2 n, and we characterize the graphs achieving equality in this bound.

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On independent domination of regular graphs

Cho

Choi

Park

2022

Journal of Graph Theory

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The domination number of a graph G $G$, denoted γ ( G ) $\gamma (G)$, is the minimum size of a dominating set of G $G$, and the independent domination number of G $G$, denoted i ( G ) $i(G)$, is the minimum size of a dominating set of G $G$ that is also independent. Let k ≥ 4 $k\ge 4$ be an integer. Generalizing a result on cubic graphs by Lam, Shiu, and Sun, we prove that i ( G ) ≤ k − 1 2 k − 1 false| V ( G ) false| $i(G)\le \frac{k-1}{2k-1}|V(G)|$ for a connected k $k$‐regular graph G $G$ that is not K k , k ${K}_{k,k}$, which is tight for k = 4 $k=4$. This answers a question by Goddard et al. in the affirmative. We also show that i ( G ) γ ( G ) ≤ k 3 − 3 k 2 + 2 2 k 2 − 6 k + 2 $\frac{i(G)}{\gamma (G)}\le \frac{{k}^{3}-3{k}^{2}+2}{2{k}^{2}-6k+2}$ for a connected k $k$‐regular graph G $G$ that is not K k , k ${K}_{k,k}$, strengthening upon a result of Knor, Škrekovski, and Tepeh. In addition, we prove that a graph G $G$ with maximum degree at most 4 satisfies i ( G ) ≤ 5 9 false| V ( G ) false| $i(G)\le \frac{5}{9}|V(G)|$, which is also tight.

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Independent Domination in Cubic Graphs

Cited by 20 publications

References 7 publications

On independent domination of regular graphs

On independent domination of regular graphs

Independent Domination in Subcubic Graphs

On independent domination of regular graphs

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