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)≤γ_e(G)≤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)=n+2 6 and γ_e(T)≤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.