Lower bounds for the domination number

Abstract

Abstract. In this note, we prove several lower bounds on the domination number of simple connected graphs. Among these are the following: the domination number is at least two-thirds of the radius of the graph, three times the domination number is at least two more than the number of cut-vertices in the graph, and the domination number of a tree is at least as large as the minimum order of a maximal matching.

On domination number and distance in graphs

“…Since, for a graph G of order n, W (G) = 1≤i<j≤n d(x i , x j ) is the Wiener index of G (see [6]) and µ(G) = 1 n(n−1) W (G) is the average distance (per definition found in [1]), it follows that γ(G) ≥ µ(G) = 1 n(n−1) W (G). As an application of our main result, we prove a conjecture in [3] by DeLaViña et al that γ(G) ≥ 1 2 (ecc G (B) + 1), where ecc G (B) denotes the eccentricity of the boundary of an arbitrary connected graph G (to be defined in Section 4). This paper is motivated by the work of Henning and Yeo in [5], where they obtained similar inequalities for total domination number γ t (rather than domination number γ).…”

Line Join Connected Domination on a Graph

“…Proof: E.Delavina [3] proved that γ(T) 3 2 l n . From theorem 3.6, we know that for any connected graph G, γ(G) ≤ γ ljcd (G).…”

“…From theorem 3.6, we know that for any connected graph G, γ(G) ≤ γ ljcd (G). Therefore, for a tree T with l leaves and n vertices γ ljcd (T) Proof: E.Delavina [3] proved that γ(G) 3 2 x . From theorem 3.6, we know that for any connected graph G, γ(G) ≤ γ ljcd (G).…”

A Note on Connected Interior Domination in Join and Corona of Two Graphs

“…These concepts are helpful to find centrally located sets to cover the entire graph. The basics definitions and theorems used in this study can be found in [2,6].…”