Vertices with the second neighborhood property in Eulerian digraphs

Abstract

The Second Neighborhood Conjecture states that every simple digraph has a vertex whose second out-neighborhood is at least as large as its first out-neighborhood, i.e. a vertex with the Second Neighborhood Property. A cycle intersection graph of an even graph is a new graph whose vertices are the cycles in a cycle decomposition of the original graph and whose edges represent vertex intersections of the cycles. By using a digraph variant of this concept, we prove that Eulerian digraphs which admit a simple cycle intersection graph have not only adhere to the Second Neighborhood Conjecture, but that local simplicity can, in some cases, also imply the existence of a Seymour vertex in the original digraph.

Cycle intersection graphs and minimum decycling sets of even graphs

“…As can be seen from the definition of a cycle intersection graph, even graphs are the only family of graphs whose entire edge set is characterized by their cycle intersection graph. However, spanning even subgraphs are a common and highly useful tool in many areas on graph theory and properties of their existence and structure in graphs and digraphs are well known [5,6]. The intention of developing this approach to find decyling sets of even graphs is that it can immediately serve as a useful tool for finding decycling sets of other graphs, possibly via either maximum spanning even subgraphs or minimum containing even graphs.…”