On the isolation numbers in graph products

Abstract

As a continuation of a previous study of isolation numbers in Cartesian and lexicographic products, we investigate isolation numbers and, more generally, $\cal F$-isolation numbers in direct, strong, lexicographic, and Cartesian products of graphs. For direct products, we derive upper bounds for the $\{K_{n_1,\ldots,n_d}\}$-isolation number in terms of isolation and total domination parameters of the factors, and establish lower bounds based on open packings. We also determine exact values for several infinite families of direct products, including $ι(P_{4\ell}\times C_{2k+1})=\ell(k+1)$. For strong products, we prove a general lower bound on $ι(G\,\boxtimes\, H,{\cal F})$ involving the $2$-packing number and provide an upper bound on $ι(G\,\boxtimes\, H)$. For lexicographic products, we determine the $\cal F$-isolation number in several general settings, obtaining exact formulas in terms of domination and total domination numbers of the first factor. Finally, for Cartesian products, we extend results from our previous work to arbitrary graph families $\cal F$. We introduce $\cal F$-isolation graphs and use $\cal F$-transversals to derive general upper bounds, together with corresponding lower bounds.