A characterization of graphs with maximum k-clique isolation number

For any positive integer k and any graph G, a subset D of vertices of G is called a k-clique isolating set of G if $G-N\left[D\right]$ does not contain k-clique as a subgraph. The k-clique isolation number of G, denoted by $\iota (G,k)$, is the minimum cardinality of a k-clique isolating set of G. Borg, Fenech and Kaemawichanurat (Discrete Math. 343 (2020) 111879) proved that if G is a connected n-vertex graph, then $\iota (G,k)\le \frac{n}{k+1}$ unless G is a k-clique, or $k=2$ and G is a 5-cycle. At the end of their paper, Borg, Fenech and Kaemawichanurat asked for a characterization of all connected n-vertex graphs G such that $\iota (G,k)=\frac{n}{k+1}$. An old result of Payan and Xuong, and independently of Fink et al., in the 1980s has already answered this problem for the case $k=1$. Very recently, the case when $k=2$ was solved by Boyer and Goddard, and the case when $k=3$ was solved by the first two authors of the present paper and Zhang. In this paper, we solve all the remaining cases. We show that except an infinite family of graphs, there are exactly 7 such graphs when $k=4$ and exactly $k+2$ such graphs when $k\ge 5$.