Some Upper Bounds for the Domination Number

Let \( G \) be a graph of order \( n(G) \), minimum degree \( \delta(G) \), diameter \( d_m(G) \), and let \( \bar{G} \) be the complement of the graph \( G \). A vertex set \( D \) is called a dominating set of \( G \), if each vertex not in \( D \) has at least one neighbor in \( D \). The domination number \( \gamma(G) \) equals the minimum cardinality of a dominating set of \( G \).
In this article we show the inequalities

1. \( \gamma(G) \leq \left\lfloor \frac{n(G)}{3} \right\rfloor, \text{ if } \delta(G) \geq 7, \)
2. \( \gamma(G) + \gamma(\bar{G}) \leq \left\lfloor \frac{n(G)}{3} \right\rfloor + 2, \text{ if } \delta(G), \delta(\bar{G}) \geq 7, \text{ and} \)
3. \( \gamma(G) \leq \left\lceil \frac{n(G)}{4} \right\rceil + 1, \text{ if } \text{dm}(G) = 2. \)

Using the concept of connectivity, we present some related upper bounds for the domination number of graphs with \( \text{dm}(G) = 2 \) and \( \text{dm}(G) = 3 \).