  A vertex in a directed graph is said to have a large second neighborhood if
it has at least as many second out-neighbors as out-neighbors. The Second
Neighborhood Conjecture, first stated by Seymour, asserts that there is a
vertex having a large second neighborhood in every oriented graph (a directed
graph without loops or digons). We prove that oriented graphs whose missing
edges can be partitioned into a (possibly empty) matching and a (possibly
empty) star satisfy this conjecture. This generalizes a result of Fidler and
Yuster. An implication of our result is that every oriented graph without a
sink and whose missing edges form a (possibly empty) matching has at least two
vertices with large second neighborhoods. This is a strengthening of a theorem
of Havet and Thomasse, who showed that the same holds for tournaments without a
sink. Moreover, we also show that the conjecture is true for oriented graphs
whose vertex set can be partitioned into an independent set and a 2-degenerate
graph.
