Nordhaus–Gaddum-Type Results on the Connected Edge Domination Number

A connected edge dominating set of a connected graph G = ( V , E ) is a subset X of E such that the edge-induced subgraph G [ X ] is connected and each e ∈ E ( G ) \ X has at least one neighbor in X . The connected edge domination number γ c ′ ( G ) of G is the minimum cardinality of a connected edge dominating set of G . An edge dominating set X of a graph G is called a 2 -edge-connected edge dominating set if G [ X ] is 2-edge-connected. The 2 -edge-connected edge dominating number γ 2 e c ′ ( G ) is the minimum size of a 2-edge-connected edge dominating set of G . In this paper, we obtain the sharp lower bounds for γ c ′ ( G ) + γ c ′ ( G ¯ ) and γ c ′ ( G ) · γ c ′ ( G ¯ ) . Moreover, we characterize the classes of graphs attaining the lower bounds and study the relationship between γ c ′ ( G ) and several other parameters, such as independent number and vertex cover number. In addition, we show that 3 ≤ γ 2 e c ′ ( G ) ≤ ⌊ 3 2 ( n - 1 ) ⌋ if G is 2-edge-connected. We also obtain the upper and lower bounds for γ 2 e c ′ ( G ) + γ 2 e c ′ ( G ¯ ) and γ 2 e c ′ ( G ) · γ 2 e c ′ ( G ¯ ) and characterize the classes of graphs attaining the lower bounds.