The Second Neighborhood Conjecture for some Special Classes of Graphs
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An oriented graph is a directed graph with no loops or digons. The second out-neighbors of a vertex u in an oriented graph are the vertices to which there exist directed paths of length 2 from u, but are not out-neighbors of u. The Second Neighborhood Conjecture, first stated by Seymour, asserts that in every oriented graph, there is a vertex which has at least as many second out-neighbors as out-neighbors. We prove that the conjecture is true for two special classes of oriented graphs. We first show that any oriented graph whose vertex set can be partitioned into an independent set and 2-degenerate graph satisfies the conjecture. Fidler and Yuster ["Remarks on the second neighborhood problem", Journal of Graph Theory, 55(3):208--220, 2007] showed that the conjecture is true for graphs obtained by removing a matching from a tournament. We improve this result by showing that for graphs that can be obtained by removing a matching from a tournament and contain no sink, there exist two vertices with as many second out-neighbors as out-neighbors.
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