On 2-Step and Hop Dominating Sets in Graphs

Abstract

Two vertices in a graph are said to 2-step dominate each other if they are at distance 2 apart. A set S of vertices in a graph G is a 2-step dominating set of G if every vertex is 2-step dominated by some vertex of S. A subset S of vertices of G is a hop dominating set if every vertex outside S is 2-step dominated by some vertex of S. The hop domination number, $\gamma _{h}(G)$, of G is the minimum cardinality of a hop dominating set of G. It is known that for a connected graph G, $\gamma _{h}(G) = |V(G)|$ if and only if G is a complete graph. We characterize the connected graphs G for which $\gamma _{h}(G) = |V(G)|-1$, which answers a question posed by Ayyaswamy and Natarajan [An. Stt. Univ. Ovidius Constanta 23(2):187–199, 2015]. We present probabilistic upper bounds for the hop domination number. We also prove that almost all graphs $G=G(n,p(n))$ have a hop dominating set of cardinality at most the total domination number if $p(n)\ll 1/n$, and almost all graphs $G=G(n,p(n))$ have a hop dominating set of cardinality at most $1+np(1+o(1))$, if p is constant. We show that the decision problems for the 2-step dominating set and hop dominating set problems are NP-complete for planar bipartite graphs and planar chordal graphs.

Appendix

In this appendix, we provide a direct proof of Theorem 11. Recall its statement.

Theorem 11

If G is a graph of order n, then $\displaystyle { \gamma _{h}(G) = \min \limits _{\mathbf{p} \, \in C^n} f(\mathbf{p}) }$.

Proof of Theorem 11

Let G be a graph with vertex set $V = \{1,2,\ldots ,n\}$. We pick randomly and independently each vertex $i \in [n]$ with probability $p_i$, where $0 \le p_i \le 1$, to form a set $X \subseteq V$. Thus, $p_i$ denotes the probability that the vertex i belongs to X; that is, $\Pr (i \in X) = p_i$ for $i \in [n]$. Let Z be the set of vertices outside X that are not at distance 2 in G from any vertex of X; that is,

$$\begin{aligned} Z = \{i \notin X \mid N_2(i) \cap X = \emptyset \}. \end{aligned}$$

The set $D = X \cup Z$ is a hop dominating set of G. Clearly, $|X|=\sum _{i=1}^n X_i$, where $X_i$ is a random variable with $X_i=1$ if $i \in X$, and $X_i=0$ otherwise. Similarly, Z can be written as a sum of n indicator random variables, say $Z_1, \ldots , Z_n$. By the linearity of expectation,

$$\begin{aligned} E(|D|) \le E(|X| + |Z|) = E(|X|) + E(|Z|). \end{aligned}$$

Hence using the well-known fact that for a random subset M of a given finite set N,

$$\begin{aligned} E(|M|) = \sum _{n \in N} \Pr (n \in M), \end{aligned}$$

we have

$$\begin{aligned} E(|X|)=\sum _{i=1}^n E(X_i)=\sum _{i=1}^np_i, \end{aligned}$$

and

$$\begin{aligned} E(|Z|)=\sum _{i=1}^n E(Z_i)=\sum _{i=1}^n (1-p_i)\prod _{j\in N_2(i)}(1-p_j). \end{aligned}$$

Thus,

$$\begin{aligned} E(|D|) \le E(|X|) + E(|Z|) \le \sum \limits _{i = 1}^n p_i \, + \, \left( \sum \limits _{i = 1}^n (1 - p_i) \cdot \prod \limits _{j \in N_2(i)} (1 - p_j) \right) = f(\mathbf{p}). \end{aligned}$$

The expectation being an average value, there is consequently a hop dominating set of G of cardinality at most E(|D|). Hence,

$$\begin{aligned} \displaystyle { \gamma _{h}(G) \le \min \limits _{\mathbf{p} \, \in C^n} f(\mathbf{p}) }. \end{aligned}$$

Now let $D^*$ be a hop dominating set of G of minimum cardinality $\gamma _{h}(G)$. Then for $\mathbf{p^*} = (p_1^*, \ldots , p_n^*)$ where $p_i^* = 1$ if $i \in D^*$ and $p_i^* = 0$ otherwise,

$$\begin{aligned} f(\mathbf{p^*}) = \sum \limits _{i = 1}^n p_i = |D^*| = \gamma _{h}(G), \end{aligned}$$

whence $\displaystyle { \gamma _{h}(G) = \min \limits _{\mathbf{p} \in C^n} f(\mathbf{p}) }$. $\square$

We show next that the result of Theorem 12 can be deduced directly from Theorem 11. Recall the statement of Theorem 12.

Theorem 12

If G is a graph of order n with $\delta _h = \delta _h(G) \ge 1$, then

$$\begin{aligned} \gamma _{h}(G) \le \left( \frac{ \ln (\delta _h + 1) + 1 }{ \delta _h + 1 } \right) n. \end{aligned}$$

Proof of Theorem 12

Following the notation introduced in the proof of Theorem 11, we let $\mathbf{p} = (p_1, \ldots , p_n)$ and we set $p_i = p$ for all $i \in [n]$, where $0 \le p \le 1$. Further, for $i \in [n]$, we let $d_i$ denote the hop-degree of the vertex i in G, and so, $d_i = d_h(i)$. Then,

$$\begin{aligned} \begin{array}{lcl} f(\mathbf{p}) &{} = &{} \displaystyle { \sum \limits _{i = 1}^n p \, + \, \sum \limits _{i = 1}^n (1 - p)^{d_i + 1} } \\ &{} \le &{} \displaystyle { np \, + \, n (1-p)^{\delta _h + 1} } \quad (\text{ since } \delta _h \le d_i\hbox { and }0 \le 1-p \le 1) \\ &{} \le &{} \displaystyle { np \, + \, n e^{-(\delta _h + 1) p} }\qquad (\text{ since } \text{ for } x \in \mathbb {R}, 1 - x \le e^{-x}). \\ \end{array} \end{aligned}$$

The function $g(p) = np \, + \, n e^{-(\delta _h + 1) p}$ is minimized when $p = p^*$ where

$$\begin{aligned} e^{-(\delta _h + 1) p^*} = \frac{1}{ \delta _h + 1 }, \end{aligned}$$

i.e., where

$$\begin{aligned} p^* = \frac{ \ln (\delta _h + 1) }{ \delta _h + 1}. \end{aligned}$$

We note that $0< p^* < 1$. Let $\mathbf{p}^* = (p^*, \ldots , p^*)$ be the n-vector each entry of which is equal to $p^*$. By Theorem 11,

$$\begin{aligned} \gamma _{h}(G) \le f(\mathbf{p}^*) \le np^* \, + \, n e^{-(\delta _h + 1) p^*} = \left( \frac{ \ln (\delta _h + 1) + 1 }{ \delta _h + 1 } \right) n. \end{aligned}$$

which is the desired upper bound. $\square$