Bipartite Theory of Graphs: Outer-Independent Domination

Abstract

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 $D \subseteq X$ is an $X$-outer-independent dominating set of $G$ if every vertex of $X \setminus D$ has an $X$-neighbor in $D$, and all vertices of $X \setminus D$ are pairwise $X$-independent. The $X$-outer-independent domination number of $G$, denoted by $\gamma _X^{oi}(G)$, is the minimum cardinality of an $X$-outer-independent dominating set of $G$. We prove several properties and bounds on the number $\gamma _X^{oi}(G)$.