Exponential Domination in Subcubic Graphs

As a natural variant of domination in graphs, Dankelmann et al. [Domination with exponential decay, Discrete Math. 309 (2009) 5877-5883] introduce exponential domination, where vertices are considered to have some dominating power that decreases exponentially with the distance, and the dominated vertices have to accumulate a sufficient amount of this power emanating from the dominating vertices. More precisely, if $S$ is a set of vertices of a graph $G$, then $S$ is an exponential dominating set of $G$ if $\sum\limits_{v\in S}\left(\frac{1}{2}\right)^{{\rm dist}_{(G,S)}(u,v)-1}\geq 1$ for every vertex $u$ in $V(G)\setminus S$, where ${\rm dist}_{(G,S)}(u,v)$ is the distance between $u\in V(G)\setminus S$ and $v\in S$ in the graph $G-(S\setminus \{ v\})$. The exponential domination number $\gamma_e(G)$ of $G$ is the minimum order of an exponential dominating set of $G$. In the present paper we study exponential domination in subcubic graphs. Our results are as follows: If $G$ is a connected subcubic graph of order $n(G)$, then $$\frac{n(G)}{6\log_2(n(G)+2)+4}\leq \gamma_e(G)\leq \frac{1}{3}(n(G)+2).$$ For every $\epsilon>0$, there is some $g$ such that $\gamma_e(G)\leq \epsilon n(G)$ for every cubic graph $G$ of girth at least $g$. For every $0<\alpha<\frac{2}{3\ln(2)}$, there are infinitely many cubic graphs $G$ with $\gamma_e(G)\leq \frac{3n(G)}{\ln(n(G))^{\alpha}}$. If $T$ is a subcubic tree, then $\gamma_e(T)\geq \frac{1}{6}(n(T)+2).$ For a given subcubic tree, $\gamma_e(T)$ can be determined in polynomial time. The minimum exponential dominating set problem is APX-hard for subcubic graphs

On exponential domination of graphs

Exponential domination in graphs evaluates the influence that a particular vertex exerts on the remaining vertices within a graph. The amount of influence a vertex exerts is measured through an exponential decay formula with a growth factor of one-half. An exponential dominating set consists of vertices who have significant influence on other vertices. In non-porous exponential domination, vertices in an exponential domination set block the influence of each other. Whereas in porous exponential domination, the influence of exponential dominating vertices are not blocked. For a graph $G,$ the non-porous and porous exponential domination numbers, denoted $\gamma_e(G)$ and $\gamma_e^*(G),$ are defined to be the cardinality of the minimum non-porous exponential dominating set and cardinality of the minimum porous exponential dominating set, respectively. This dissertation focuses on determining lower and upper bounds of the non-porous and porous exponential domination number of the King grid $\mathcal{K}_n,$ Slant grid $\mathcal{S}_n,$ $n$-dimensional hypercube $Q_n,$ and the general consecutive circulant graph $C_{n,[\ell]}.$ A method to determine the lower bound of the non-porous exponential domination number for any graph is derived. In particular, a lower bound for $\gamma_e^*(Q_n)$ is found. An upper bound for $\gamma_e^*(Q_n)$ is established through exploiting distance properties of $Q_n.$ For any grid graph $G,$ linear programming can be incorporated with the lower bound method to determine a general lower bound for $\gamma_e^*(G).$ Applying this technique to the grid graphs $\mathcal{K}_n$ and $\mathcal{S}_n$ yields lower bounds for $\gamma_e^*(\mathcal{K}_n)$ and $\gamma_e^*(\mathcal{S}_n).$ Upper bound constructions for $\gamma_e^*(\mathcal{K}_n)$ and $\gamma_e^*(\mathcal{S}_n)$ are also derived. Finally, it is shown that $\gamma_e(C_{n,[\ell]}) = \gamma_e^*(C_{n,[\ell]}).

Exponential Independence in Subcubic Graphs

A set $S$ of vertices of a graph $G$ is exponentially independent if, for every vertex $u$ in $S$, $$\sum\limits_{v\in S\setminus \{ u\}}\left(\frac{1}{2}\right)^{{\rm dist}_{(G,S)}(u,v)-1}<1,$$ where ${\rm dist}_{(G,S)}(u,v)$ is the distance between $u$ and $v$ in the graph $G-(S\setminus \{ u,v\})$. The exponential independence number $\alpha_e(G)$ of $G$ is the maximum order of an exponentially independent set in $G$. In the present paper we present several bounds on this parameter and highlight some of the many related open problems. In particular, we prove that subcubic graphs of order $n$ have exponentially independent sets of order $\Omega(n/\log^2(n))$, that the infinite cubic tree has no exponentially independent set of positive density, and that subcubic trees of order $n$ have exponentially independent sets of order $(n+3)/4$