Further results on the number of cliques in graphs covered by long cycles

Let $\mathrm{\Gamma }(n,k)$ be the set of $2$-connected $n$-vertex graphs containing an edge that is not on any cycle of length at least $k+1.$ Let $g_s(n,k)$ denote the maximum number of $s$-cliques in a graph in $\mathrm{\Gamma }(n,k)\mathrm{.}$ Recently, Ji and Ye [SIAM J. Discrete Math., 37 (2023) 917-924] determined $g_s(n,k)\mathrm{.}$ They remark that it is interesting to characterize the extremal graphs. In this paper, we give such a characterization.