Relating domination, exponential domination, and porous exponential domination

The domination number γ(G) of a graph G, its exponential domination number γe(G), and its porous exponential domination number γe∗(G) satisfy γe∗(G)≤γe(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 γe∗(G), we are led to the definition of the fractional porous exponential domination number γe,f∗(G) of a graph G. For a subcubic tree T of order n, we show γe,f∗(T)=n+26 and γe(T)≤2γe,f∗(T). We characterize the two classes of subcubic trees T with γe(T)=γe,f∗(T) and γ(T)=γe(T), respectively. Using linear programming arguments, we establish several lower bounds on the fractional porous exponential domination number in more general settings.