On externally supported independence number of graphs

Abstract

We introduce the \emph{externally supported independence number} $\hat{I}{\pm }_{\mathrm{es}}(G)$ 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 ${\hat{I}}^1(G)$. We show that computing $\hat{I}{\pm }_{\mathrm{es}}(G)$ is NP-hard, while for trees we present a linear-time algorithm. We also establish several sharp bounds on $\hat{I}{\pm }_{\mathrm{es}}(G)$ for general graphs, with additional refined results for trees. In several cases, we completely describe the extreme graph classes attaining these bounds.

22 pages, 4 figures, one algorithm