Diameter of paired domination edge-critical graphs

A paired dominating set of a graph G without isolated vertices is a dominating set of G whose induced subgraph has a perfect matching. The paired domination number γpr(G) of G is the minimum cardinality amongst all paired dominating sets of G. The graph G is paired domination edge-critical (γprEC) if for every e ∈ E(G), γpr(G+ e) < γpr(G). We investigate the diameter of γprEC graphs. To this effect we characterize γprEC trees. We show that for arbitrary even k ≥ 4 there exists a kprEC graph with diameter two. We provide an example which shows that the maximum diameter of a kprEC graph is at least k− 2 and prove that it is at most min{2k − 6, 3k/2 + 3}. ∗ Recipient of an Undergraduate Student Research Award from the Canadian National Science and Engineering Research Council, Summer 2006. Current address: Department of Mathematics, Simon Fraser University, Burnaby, BC, Canada V5A1S6. [email protected] † Research supported in part by the South African National Research Foundation and the University of KwaZulu-Natal. ‡ Research supported by the Natural Sciences and Engineering Research Council of Canada. 280 EDWARDS, GIBSON, HENNING AND MYNHARDT