Bipartite Theory of Graphs: Outer-Independent Domination

Let G = ( V , E ) be a bipartite graph with partite sets X and Y . Two vertices of X are X -adjacent if they have a common neighbor in Y , and they are X -independent otherwise. A subset D ⊆ X is an X -outer-independent dominating set of G if every vertex of X \ D has an X -neighbor in D , and all vertices of X \ D are pairwise X -independent. The X -outer-independent domination number of G , denoted by γ X o i ( G ) , is the minimum cardinality of an X -outer-independent dominating set of G . We prove several properties and bounds on the number γ X o i ( G ) .