The Seymour’s Second Neighborhood Conjecture (SSNC) asserts that every oriented graph \(G\) contains a vertex \(v\) such that \(|N^{++}(v)| \ge |N^+(v)|\) . Such a vertex is said to have the second neighborhood property. The conjecture was first proved for tournaments by Fisher and later Havet and Thomassé provided a different proof using the median order approach. Following this approach, we give another simple proof that the SSNC holds for a tournament missing a matching and a tournament missing a star. Let s, t be two non-negative integers. Denote \(\mathcal {D}_{s,t}\) the class of oriented graphs whose vertex set has a partition \((A,B)\) such that the subgraph induced by \(A\) is s-degenerate and the subgraph induced by \(B\) is t-degenerate. In this paper, we show that the SSNC holds for oriented graphs in \(\mathcal {D}_{0,3}\cup \mathcal {D}_{1,1}\) . In [3], Dara, Francis, Jacob and Narayanan showed that the conjecture holds for oriented graphs in \(\mathcal {D}_{0,2}\) . Thus, we extend their result.
