Abstract
A dominating set D is a weakly connected dominating set of a connected graph G = ( V , E ) if ( V , E ∩ ( D × V ) ) is connected. The weakly connected domination number of G , denoted γ wc ( G ) , is min { | S | | S is a weakly connected dominating set of G } . We characterize graphs G for which γ ( H ) = γ wc ( H ) for every connected induced subgraph H of G , where γ is the domination number of a graph. We provide a constructive characterization of trees T for which γ ( T ) = γ wc ( T ) . Lastly, we constructively characterize the trees T in which every vertex belongs to some weakly connected dominating set of cardinality γ wc ( T ) .
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