A contribution to the second neighborhood problem

2011

Seymour's Second Neighborhood Conjecture asserts that every digraph (without digons) has a vertex whose first out-neighborhood is at most as large as its second out-neighborhood. We prove its weighted version for tournaments missing a generalized star. As a consequence the weighted version holds for tournaments missing a sun, star, or a complete graph.

2011

Seymour's Second Neighborhood Conjecture asserts that every digraph (without digons) has a vertex whose first out-neighborhood is at most as large as its second out-neighborhood. We prove its weighted version for tournaments missing a generalized star. As a consequence the weighted version holds for tournaments missing a sun, star, or a complete graph.

Seymour's second neighborhood conjecture states that every simple digraph (without digons) has a vertex whose first out-neighborhood is at most as large as its second out-neighborhood. Such a vertex is said to have the second neighborhood property (SNP). We define "good" digraphs and prove a statement that implies that every feed vertex of a tournament has the SNP. In the case of digraphs missing a matching, we exhibit a feed vertex with the SNP by refining a proof due to Fidler and Yuster and using good digraphs. Moreover, in some cases we exhibit two vertices with SNP.

2011

Let $D$ be a digraph without digons. Seymour's second neighborhood conjecture states that $D$ has a vertex $v$ such that $d^+(v)\leq d^{++}(v)$. Under some conditions, we prove this conjecture for digraphs missing $n$ disjoint stars. Weaker conditions are required when $n=2$ or 3. In some cases we exhibit 2 such vertices.

arXiv: Discrete Mathematics, 2018

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-degenerat...

Involve, a Journal of Mathematics, 2009

Let D be a simple digraph without loops or digons. For any v ∈ V (D) let N 1 (v) be the set of all nodes at out-distance 1 from v and let N 2 (v) be the set of all nodes at out-distance 2. We show that if the underlying graph is triangle-free, there must exist some v ∈ V (D) such that |N 1 (v)| ≤ |N 2 (v)|. We provide several properties a "minimal" graph which does not contain such a node must have. Moreover, we show that if one such graph exists, then there exist infinitely many. Conjecture 1.1 (Seymour's second neighborhood conjecture). Let D be a directed graph. Then there exists a vertex v 0 ∈ V (D) such that |N 1 (v 0)| ≤ |N 2 (v 0)|. Dean and Latka [1995] conjectured this to be true when D is a tournament. Dean's conjecture was subsequently proven by Fisher [1996]. Further, Kaneko and Locke [2001] showed Conjecture 1.1 to be true if the minimum out-degree of vertices in D is less than 7, while Cohn, Wright and Godbole [Cohn et al. 2009] MSC2000: 05C20.

Arxiv preprint arXiv: …, 2008

Let D be a simple digraph without loops or digons. For any v ∈ V (D) let N 1 (v) be the set of all nodes at out-distance 1 from v and let N 2 (v) be the set of all nodes at out-distance 2. We provide conditions under which there must exist some v ∈ V (D) such that |N 1 (v)| ≤ |N 2 (v)|, as well as examine extremal properties in a minimal graph which does not have such a node. We show that if one such graph exists, then there exist infinitely many.

arXiv (Cornell University), 2023

For a vertex x of a digraph, d + (x) (d − (x), resp.) is the number of vertices at distance 1 from (to, resp.) x and d ++ (x) is the number of vertices at distance 2 from x. In 1995, Seymour conjectured that for any oriented graph D there exists a vertex x such that d + (x) ≤ d ++ (x). In 2006, Sullivan conjectured that there exists a vertex x in D such that d − (x) ≤ d ++ (x). We give a sufficient condition in terms of the number of transitive triangles for an oriented graph to satisfy Sullivan's conjecture. In particular, this implies that Sullivan's conjecture holds for all orientations of planar graphs and of triangle-free graphs. An oriented graph D is an oriented split graph if the vertices of D can be partitioned into vertex sets X and Y such that X is an independent set and Y induces a tournament. We also show that the two conjectures hold for some families of oriented split graphs, in particular, when Y induces a regular or an almost regular tournament.

arXiv (Cornell University), 2016

Seymour's Second Neighborhood Conjecture asserts that every oriented graph has a vertex whose first out-neighborhood is at most as large as its second out-neighborhood. Combs are the graphs having no induced C4, C4, C5, chair or chair. We characterize combs using dependency digraphs. We characterize the graphs having no induced C4, C4, chair or chair using dependency digraphs. Then we prove that every oriented graph missing a comb satisfies this conjecture. We then deduce that every oriented comb and every oriented threshold graph satisfies Seymour's conjecture.

2011

Seymour's Second Neighborhood Conjecture asserts that every digraph (without digons) has a vertex whose first out-neighborhood is at most as large as its second out-neighborhood. It is proved for tournaments, tournaments missing a matching and tournaments missing a generalized star. We prove this conjecture for classes of digraphs whose missing graph is a comb, a complete graph minus 2 independent edges, or a complete graph minus the edges of a cycle of length 5.