On externally supported independence number of graphs

We introduce the \emph{externally supported independence number} ${\alpha }_{\mathrm{e}\mathrm{s}}(G)$ of a graph $G$ as the maximum cardinality of an independent set $B$ with an additional condition, that vertices from $N(B)$ are dominated by vertices in $V(G)-N[B]$. This parameter yields an improved upper bound on the isolation number $\iota (G)$. We show that computing ${\alpha }_{\mathrm{e}\mathrm{s}}(G)$ is NP-hard, while for trees we present a linear-time algorithm. We also establish several sharp bounds on ${\alpha }_{\mathrm{e}\mathrm{s}}(G)$ for general graphs, with additional refined results for trees. In several cases, we completely describe the extreme graph classes attaining these bounds.