Corpus ID: 210164677

Towards the Small Quasi-Kernel Conjecture

  title={Towards the Small Quasi-Kernel Conjecture},
  author={Alexandr V. Kostochka and Ruth Luo and Songling Shan},
  journal={arXiv: Combinatorics},
Let $D=(V,A)$ be a digraph. A vertex set $K\subseteq V$ is a quasi-kernel of $D$ if $K$ is an independent set in $D$ and for every vertex $v\in V\setminus K$, $v$ is at most distance 2 from $K$. In 1974, Chv\'atal and Lov\'asz proved that every digraph has a quasi-kernel. P. L. Erd\H{o}s and L. A. Sz\'ekely in 1976 conjectured that if every vertex of $D$ has a positive indegree, then $D$ has a quasi-kernel of size at most $|V|/2$. This conjecture is only confirmed for narrow classes of digraphs… Expand
Kernels and Small Quasi-Kernels in Digraphs
A directed graph $D=(V(D),A(D))$ has a kernel if there exists an independent set $K\subseteq V(D)$ such that every vertex $v\in V(D)-K$ has an ingoing arc $u\mathbin{\longrightarrow}v$ for some $u\inExpand
Algorithmic aspects of quasi-kernels
It is shown that, not only sink-free digraphs occasionally fail to contain two disjoint quasi-kernels, but it is computationally hard to distinguish those that do from those that don't, and it is proved that the problem of computing a small quasikernel is polynomial time solvable for orientations of trees but is computationationally hard in most other cases. Expand
In a digraph, a quasi-kernel is a subset of vertices that is independent and such that every vertex can reach some vertex in that set via a directed path of length at most two. Whereas Chvátal andExpand
Problems that I would like Somebody to Solve
3 Some Smaller Problems 5 3.1 Slow Tribonacci Walks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3.2 An Adversarial Chernoff Bound . . . . . . . . . . . . . . . . . . . . . . . .Expand


Disjoint quasi-kernels in digraphs
A quasi-kernel in a digraph is an independent set of vertices such that any vertex in the digraph can reach some vertex in the set via a directed path of length at most two. Chvatal and Lovasz provedExpand
On the number of quasi-kernels in digraphs
In 1974, Chvátal and Lovász proved that every digraph has a vertex set D 1⁄4 (V, A), and it is shown that for every v 2 V X there exists x 2 X such that vx 2 A. Expand
Every Planar Map Is Four Colorable
As has become standard, the four color map problem will be considered in the dual sense as the problem of whether the vertices of every planar graph (without loops) can be colored with at most fourExpand
On weakly ordered systems
The statement "x>y" may be read "x dominates y." Transitivity is not assumed ; a transitive weakly ordered system is a partially ordered system. By a solution of a weakly ordered system is meant aExpand
About quasi-kernels in a digraph
It is proved that every graph without kernel has at least three distinct quasi-kernels. Expand
Planar kernel and grundy with d≤3, dout≤2, din≤2 are NP-complete
  • A. Fraenkel
  • Computer Science, Mathematics
  • Discret. Appl. Math.
  • 1981
Abstract It is proved that the questions whether a finite diagraph G has a kernel K or a Sprague—Grundy function g are NP-complete even if G is a cyclic planar digraph with degree constraints d out (Expand
Every planar map is four colorable. Part II: Reducibility
On the computational complexity of finding a kernel
  • Report No. CRM- 300, Centre de Recherches Mathématiques, Université de Montréal,
  • 1973
Every directed graph has a semi-kernel