Domination versus independent domination in regular graphs

Knor, Martin; Škrekovski, Riste; Tepeh, Aleksandra

doi:10.1002/jgt.22711

Cited by 1 publication

(2 citation statements)

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“…Babikir and Henning [3] showed that the statement of Goddard et al in the previous paragraph also holds for k-regular graphs when ∈ k {4, 5, 6}. Very recently, Knor, Škrekovski, and Tepeh [12] generalized the result to all ≥ k 3; namely, it is now known that for all…”

Section: Introductionmentioning

confidence: 84%

“…Babikir and Henning [3] showed that the statement of Goddard et al in the previous paragraph also holds for

k

‐regular graphs when

k\in \{4,5,6\}

. Very recently, Knor, Škrekovski, and Tepeh [12] generalized the result to all

k\ge 3

; namely, it is now known that for all

k\ge 3

, if

G

is a connected

k

‐regular graph, then

\frac{i(G)}{\gamma (G)}\le \frac{k}{2}

, and equality holds if and only if

G={K}_{k,k}

.…”

Section: Introductionmentioning

confidence: 98%

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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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Section: Introductionmentioning

confidence: 84%

“…Babikir and Henning [3] showed that the statement of Goddard et al in the previous paragraph also holds for