Distance domination-critical graphs

Abstract A set D of vertices in a connected graph G is called a k -dominating set if every vertex in G − D is within distance k from some vertex of D . The k -domination number of G , γ k ( G ) , is the minimum cardinality over all k -dominating sets of G . A graph G is k -distance domination-critical if γ k ( G − x ) γ k ( G ) for any vertex x in G . This work considers properties of k -distance domination-critical graphs and establishes a best possible upper bound on the diameter of a 2-distance domination-critical graph G , that is, d ( G ) ≤ 3 ( γ 2 − 1 ) for γ 2 ≥ 2 .