Oriented Graph Saturation

For graphs \( G \) and \( H \), \( H \) is said to be \( G \)-saturated if it does not contain a subgraph isomorphic to \( G \), but for any edge \( e \in H^c \), the complement of \( H \), \( H + e \), contains a subgraph isomorphic to \( G \). The minimum number of edges in a \( G \)-saturated graph on \( n \) vertices is denoted \( \text{sat}(n, G) \). While digraph saturation has been considered with the allowance of multiple arcs and \(2\)-cycles, we address the restriction to oriented graphs. First, we prove that for any oriented graph \( D \), there exist \( D \)-saturated oriented graphs, and hence show that \( \text{sat}(n, D) \), the minimum number of arcs in a \( D \)-saturated oriented graph on \( n \) vertices, is well defined for sufficiently large \( n \). Additionally, we determine \( \text{sat}(n, D) \) for some oriented graphs \( D \), and examine some issues unique to oriented graphs.