On the Relationships between Domination, Isolation, and Packing

We consider the relationships between the domination number of graph, denoted $\gamma$, and the distance-$2$ domination number, denoted ${\gamma }_2$, and three parameters that lie between them: the packing number, denoted $\rho$, the lower packing number, denoted ${\rho }_L$, and the isolation number, denoted $\iota$. There has been recent attention on the question of whether $\gamma \mathrm{/}\rho$ is bounded or unbounded for various families of graphs. We consider similar questions for the ratios of the five parameters. In particular we show that, while $\gamma \mathrm{/}{\rho }_L$ is unbounded in trees, it holds that $\iota \mathrm{/}{\gamma }_2$ is less than $2$ for all trees. Further, $\gamma \mathrm{/}{\rho }_L$ is at most $3$ in interval graphs, at most~$4$ in permutation graphs, and at most $5$ in general asteroidal-triple-free graphs. We also show that every tree has a set of vertices that is both isolating and a packing, and characterize trees where $\rho ={\rho }_L$.