Relating Domination, Exponential Domination, and Porous Exponential Domination

The domination number $γ(G)$ of a graph $G$, its exponential domination number $γ_e(G)$, and its porous exponential domination number $γ_e^*(G)$ satisfy $γ_e^*(G)\leq γ_e(G)\leq γ(G)$. We contribute results about the gaps in these inequalities as well as the graphs for which some of the inequalities hold with equality. Relaxing the natural integer linear program whose optimum value is $γ_e^*(G)$, we are led to the definition of the fractional porous exponential domination number $γ_{e,f}^*(G)$ of a graph $G$. For a subcubic tree $T$ of order $n$, we show $γ_{e,f}^*(T)=\frac{n+2}{6}$ and $γ_e(T)\leq 2γ_{e,f}^*(T)$. We characterize the two classes of subcubic trees $T$ with $γ_e(T)=γ_{e,f}^*(T)$ and $γ(T)=γ_e(T)$, respectively. Using linear programming arguments, we establish several lower bounds on the fractional porous exponential domination number in more general settings.

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