On the C ₄ -isolation number of a graph

Let [Formula: see text] be the cycle of length [Formula: see text]. For any graph [Formula: see text], a set [Formula: see text] of vertices of [Formula: see text] is a [Formula: see text]-isolating set of [Formula: see text] if the graph obtained from [Formula: see text] by deleting the closed neighbourhood of [Formula: see text] contains no [Formula: see text] as a subgraph. The [Formula: see text]-isolation number of [Formula: see text], denoted by [Formula: see text], is the cardinality of a smallest [Formula: see text]-isolating set of [Formula: see text]. Borg (2020) and Borg et al. (2022) proved that if [Formula: see text] is a connected graph of order [Formula: see text] and size [Formula: see text], then [Formula: see text] and [Formula: see text]. Recently, Bartolo, Borg and Scicluna (2024) showed that if [Formula: see text] is a connected graph of order [Formula: see text] that is not one of nine determined graphs, then [Formula: see text]. In this paper, we prove that if [Formula: see text] is a connected graph of size [Formula: see text], then [Formula: see text], and we characterize all graphs that attain the bound. For general [Formula: see text], it is worth noting that the problem on [Formula: see text]-isolation remains open.