Relating Domination, Exponential Domination, and Porous Exponential Domination

TL;DR

This paper explores the relationships and inequalities among various domination numbers in graphs, introduces a fractional version of porous exponential domination, and characterizes specific classes of trees where these bounds are tight.

Contribution

It defines the fractional porous exponential domination number and characterizes trees where equality holds, advancing understanding of domination parameters.

Findings

For subcubic trees, mma_{e,f}^*(T) = (n+2)/6.

Bounds on mma_e(T) in terms of mma_{e,f}^*(T).

Characterization of trees with equality in domination parameters.

Abstract

The domination number $\gamma(G)$ of a graph $G$, its exponential domination number $\gamma_e(G)$, and its porous exponential domination number $\gamma_e^*(G)$ satisfy $\gamma_e^*(G)\leq \gamma_e(G)\leq \gamma(G)$. 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 $\gamma_e^*(G)$, we are led to the definition of the fractional porous exponential domination number $\gamma_{e,f}^*(G)$ of a graph $G$. For a subcubic tree $T$ of order $n$, we show $\gamma_{e,f}^*(T)=\frac{n+2}{6}$ and $\gamma_e(T)\leq 2\gamma_{e,f}^*(T)$. We characterize the two classes of subcubic trees $T$ with $\gamma_e(T)=\gamma_{e,f}^*(T)$ and $\gamma(T)=\gamma_e(T)$, respectively. Using linear programming arguments, we establish several lower bounds on the fractional porous exponential domination number in more general settings.