Robust domination in random graphs

Abstract

In this paper, we study “robust” dominating sets of random graphs that retain the domination property even if a small deterministic set of edges are removed. We motivate our study by illustrating with examples from wireless networks in harsh environments. We then use the probabilistic method and martingale difference techniques to determine sufficient conditions for the asymptotic optimality of the robust domination number. We also discuss robust domination in sparse random graphs where the number of edges grows at most linearly in the number of vertices.

Appendix

Standard Deviation Estimate: Let $Z_i, 1 \le i \le t$ be independent Bernoulli random variables satisfying ${\mathbb {P}}(Z_i = 1) = p_i = 1-{\mathbb {P}}(Z_i = 0).$ If $W_t = \sum _{i=1}^{t} Z_i$ and $\mu _t = {\mathbb {E}}W_t,$ then for any $0< \eta < \frac{1}{2}$ we have that

$$\begin{aligned} {\mathbb {P}}\left( \left| W_t-\mu _t\right| \ge \eta \mu _t\right) \le 2\exp \left( -\frac{\eta ^2}{4}\mu _t\right) . \end{aligned}$$

(34)

For a proof of (34), we refer to Corollary $A.1.14,$ pp. $312,$ Alon and Spencer (2008).

$\underline{Montonicity of u_n(x)}$: The function $u_n(x):= \frac{\log (nx)}{|\log (1-x)|}$ has a derivative

$$\begin{aligned} u'_n(x) = \frac{H(x)-x\log {n}}{x(1-x)|\log (1-x)|^2} \end{aligned}$$

where

$$\begin{aligned} H(x):= - x \cdot \log {x} -(1-x) \cdot \log (1-x) \end{aligned}$$

(35)

is the binary entropy function and logarithms are natural throughout. If $x > \frac{1}{2},$ then $H(x) - x\log {n} \le 1-\frac{\log {n}}{2} <0$ for all $n \ge 4.$ The numerator $H(x)-x\log {n}$ has derivative $\log \left( \frac{1}{x}-1\right) - \log {n} < 0$ for all $x > \frac{1}{n+1}.$ Thus for $\frac{\lambda }{n}< x < \frac{1}{2}$ and $\lambda > 1$ we use $(1-x)|\log (1-x)| <x$ to get that $H(x)-x\log {n}$ is bounded above by

$$\begin{aligned} H\left( \frac{\lambda }{n}\right) - \frac{\lambda \log {n}}{n} = \frac{-\lambda \log {\lambda }}{n} - \left( 1-\frac{\lambda }{n}\right) \log \left( 1-\frac{\lambda }{n}\right) \le -\frac{\lambda \log {\lambda }}{n} + \frac{\lambda }{n} \end{aligned}$$

which is strictly less than zero if $\lambda > e.$