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Published online by Cambridge University Press: 13 September 2024
A cycle C of a graph G is dominating if $V(C)$ is a dominating set and
$V(G)\backslash V(C)$ is an independent set. Wu et al. [‘Degree sums and dominating cycles’, Discrete Mathematics 344 (2021), Article no. 112224] proved that every longest cycle of a k-connected graph G on
$n\geq 3$ vertices with
$k\geq 2$ is dominating if the degree sum is more than
$(k+1)(n+1)/3$ for any
$k+1$ pairwise nonadjacent vertices. They also showed that this bound is sharp. In this paper, we show that the extremal graphs G for this condition satisfy
$(n-2)/3K_1\vee (n+1)/3K_2 \subseteq G \subseteq K_{(n-2)/3}\vee (n+1)/3K_2$ or
$2K_1\vee 3K_{(n-2)/3}\subseteq G \subseteq K_2\vee 3K_{(n-2)/3}.$
This research was supported by NSFC under grant number 12101324, by NJUPT under grant number NY221025 and by Foundation of Jiangsu Provincial Double-Innovation Doctor Program under grant number JSSCBS20210533.