# A Contribution to the Second Neighborhood Problem

## Related Articles

- UNDERLYING GRAPHS OF 3-QUASI-TRANSITIVE DIGRAPHS AND 3-TRANSITIVE DIGRAPHS RUIXIA WANG, SHIYING WANG. RUIXIA WANG; SHIYING WANG // Discussiones Mathematicae: Graph Theory;2013, Vol. 33 Issue 2, p429
A digraph is 3-quasi-transitive (resp. 3-transitive), if for any path x0x1 x2x3 of length 3, x0 and x3 are adjacent (resp. x0 dominates x3). CÃ©sar HernÃ¡ndez-Cruz conjectured that if D is a 3-quasi-transitive digraph, then the underlying graph of D, UG(D), admits a 3-transitive orientation....

- On Sullivan's conjecture on cycles in 4-free and 5-free digraphs. Liang, Hao; Xu, Jun // Acta Mathematica Sinica;Jan2013, Vol. 29 Issue 1, p53
For a simple digraph G, let Î²( G) be the size of the smallest subset X âŠ† E( G) such that Gâˆ’X has no directed cycles, and let Î³( G) be the number of unordered pairs of nonadjacent vertices in G. A digraph G is called k-free if G has no directed cycles of length at most k. This...

- SOME REMARKS ON THE STRUCTURE OF STRONG k-TRANSITIVE DIGRAPHS. HERNÁNDEZ-CRUZ, CÉSAR; MONTELLANO-BALLESTEROS, JUAN JOSÉ // Discussiones Mathematicae: Graph Theory;2014, Vol. 34 Issue 4, p651
A digraph D is k-transitive if the existence of a directed path (v0, v1, ..., vk), of length k implies that (v0, vk) âˆˆ & A(D). Clearly, a 2-transitive digraph is a transitive digraph in the usual sense. Transitive digraphs have been characterized as compositions of complete digraphs on an...

- Characterization of Line Sidigraphs. Sampathkumar, E.; Subramanya, M. S.; Reddy, P. Siva Kota // Southeast Asian Bulletin of Mathematics;2011, Vol. 35 Issue 2, p297
A sigraph (sidigraph) is an ordered pair S = (G; Ïƒ) (S = (D; Ïƒ)), where G = (V, E) (D = (V, A) is a graph (digraph) called the underlying graph (underlying digraph) of S and Ïƒ : E â†’ {+, -} (&sigma: A â†’ {+, -}) is a function. The line sigraph (line sidigraph) of a sigraph...

- Rainbow Connection in 3-Connected Graphs. Li, Xueliang; Shi, Yongtang // Graphs & Combinatorics;Sep2013, Vol. 29 Issue 5, p1471
An edge-colored graph G is rainbow connected if any two vertices are connected by a path whose edges have distinct colors. The rainbow connection number of a connected graph G, denoted by rc( G), is the smallest number of colors that are needed in order to make G rainbow connected. In this...

- Neighborhood Unions for the Existence of Disjoint Chorded Cycles in Graphs. Gao, Yunshu; Li, Guojun; Yan, Jin // Graphs & Combinatorics;Sep2013, Vol. 29 Issue 5, p1337
A chorded cycle is a cycle with at least one chord. We prove that if G is a simple graph with order n â‰¥ 4 k and $${|N_G(x)\cup N_G(y)|\geq 4k+1}$$ for each nonadjacent pair of vertices x and y, then G contains k vertex-disjoint chorded cycles. The degree condition is sharp in general.

- $$\varPi $$ -Kernels in Digraphs. Galeana-Sánchez, Hortensia; Montellano-Ballesteros, Juan // Graphs & Combinatorics;Nov2015, Vol. 31 Issue 6, p2207
Let $$D=(V(D), A(D))$$ be a digraph, $$DP(D)$$ be the set of directed paths of $$D$$ and let $$\varPi $$ be a subset of $$DP(D)$$ . A subset $$S\subseteq V(D)$$ will be called $$\varPi $$ -independent if for any pair $$\{x, y\} \subseteq S$$ , there is no $$xy$$ -path nor $$yx$$ -path in...

- BROKEN CIRCUITS IN MATROIDS--DOHMEN'S INDUCTIVE PROOF. KORDECKI, WOJCIECH; ŁYCZKOWSKA-HANĆKOWIAK, ANNA // Discussiones Mathematicae: Graph Theory;2013, Vol. 33 Issue 3, p599
Dohmen [4] gives a simple inductive proof of Whitney's famous broken circuits theorem. We generalise his inductive proof to the case of matroids.

- Path-Bicolorable Graphs. Brandst�dt, Andreas; Golumbic, Martin; Le, Van; Lipshteyn, Marina // Graphs & Combinatorics;Nov2011, Vol. 27 Issue 6, p799
In this paper, we introduce the notion of path-bicolorability that generalizes bipartite graphs in a natural way: For k = 2, a graph G = ( V, E) is P -bicolorable if its vertex set V can be partitioned into two subsets (i.e., color classes) V and V such that for every induced P (a path with...