Characterization of Paired Domination Number of a Graph

Paired domination is a relatively interesting concept introduced by Teresa W. Haynes [9] recently with the following application in mind. If we think of each vertex s ∈ S, as the location of a guard capable of protecting each vertex dominated by S, then for a paired domination the guards location must be selected as adjacent pairs of vertices so that each guard is assigned one other and they are designated as a backup for each other. A set S V is a paired dominating set if S is a dominating set of G and the induced subgraph has a perfect matching. The paired domination number pr(G) is the minimum cardinality taken over all paired dominating sets in G. The minimum number of colours required to colour all the vertices so that adjacent vertices do not receive the same colour and is denoted by . In [3], Mahadevan G proved that pr +   2n 1, and characterized the corresponding extremal graphs of order up to 2n 5. In this paper we characterize the classes of all graphs whose sum of paired domination number and chromatic number equals to 2n 6, for any n ≥ 4.