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Almost all digraphs have a kernel

Ioan Tomescu
1990 Discrete Mathematics  
In this paper it is shown that almost all digraphs G of order n contain only kernels K such that logn-loglogn-1.43~IKJ~logn-Ioglogn+2.11 and the number K(G) of kernels of G verifies na.9i3+o(i) < K(G)  ...  < h'+O(') as Il+ m, This gives a positive answer to a conjecture raised by P.  ...  This result answers affirmatively to the following conjecture raised by Hansen [2] : Almost all digraphs on n vertices have a kernel whenever n + 03.  ... 
doi:10.1016/0012-365x(90)90373-p fatcat:xjjhuzqqkvayxo622oozpuohwe

Generating Functions For Kernels of Digraphs (Enumeration & Asymptotics for Nim Games) [article]

Cyril Banderier, Vlady Ravelomanana
2004 arXiv   pre-print
) in order to get exact and asymptotic results, e.g. for the existence of a kernel in a circuit or in a unicircuit digraph.  ...  This is a first step toward a generatingfunctionology treatment of kernels, while using, e.g., an approach "a la Wright".  ...  We also thank an anomymous referee for detailed comments on a somehow preliminary version of this work.  ... 
arXiv:math/0411138v1 fatcat:z6qkbcfuwjfdpaz2hlmxixj2xa

Independent and monochromatic absorbent sets in infinite digraphs

Alejandro Contreras-Balbuena, Hortensia Galeana-Sánchez, Rocío Rojas-Monroy
2015 AKCE International Journal of Graphs and Combinatorics  
A set AN ⊂ V (D) is an A-kernel if it satisfies the following conditions:  ...  Let D be a digraph, we say that it is an m-coloured digraph if the arcs of D are coloured with at most m-colours. is also an arc of D.  ...  Almost all of the historical conditions for kernels and kernels by monochromatic paths are for finite digraphs, in the case of infinite digraphs the main concern is the possible existence of certain structures  ... 
doi:10.1016/j.akcej.2015.11.005 fatcat:iggstwj3bzdvhb6wdubn64vipm

Page 3568 of Mathematical Reviews Vol. , Issue 91G [page]

1991 Mathematical Reviews  
Qiao Li (PRC-HEF) 91g:05060 05C20 Tomescu, Ioan (R-BUCH) Almost all digraphs have a kernel. Discrete Math. 84 (1990), no. 2, 181-192.  ...  Summary: “We show that almost all digraphs G of order n contain only kernels K such that logn —loglogn — 1.43 < |K| < logn — loglogn + 2.11 and the number K(G) of kernels of G satisfies n-913+0(1) < K(  ... 

New classes of panchromatic digraphs

Hortensia Galeana-Sánchez, Micael Toledo
2015 AKCE International Journal of Graphs and Combinatorics  
A digraph D = (V, A) with a k-colouring of its arcs ς : A → [k] is said to have a ς-kernel if there exists a subset K of V such that there are no monochromatic uv-paths for any two vertices u, v ∈ K ,  ...  D is said to be a panchromatic digraph if, for every k ≤ |A| and every k-colouring ς : A → [k], D has a ς-kernel. In this paper we study the panchromaticity of cycles.  ...  In that context, a kernel represents a set of winning positions in a game. Almost forty years later, in 1982, the concept of ς -kernel makes its first appearance in [4] .  ... 
doi:10.1016/j.akcej.2015.11.006 fatcat:r47zl6wnbzaspav2yf3r7nwwea

Page 6631 of Mathematical Reviews Vol. , Issue 92m [page]

1992 Mathematical Reviews  
A graph is an R~-digraph if every proper induced subdigraph has a kernel but the whole graph does not have a kernel.  ...  (MEX-NAM) Extending kernel perfect digraphs to kernel perfect critical digraphs. Discrete Math. 94 (1991), no. 3, 181-187.  ... 

Perfect graphs with polynomially computable kernels [article]

Adèle Pass-Lanneau, Ayumi Igarashi, Frédéric Meunier
2018 arXiv   pre-print
Not all digraphs have a kernel, but a theorem due to Boros and Gurvich guarantees the existence of a kernel in every clique-acyclic orientation of a perfect graph.  ...  However, an open question is the complexity status of the computation of a kernel in such a digraph.  ...  Not all digraphs admit a kernel: for example a directed cycle of length three has no kernel. It has been shown by Chvátal [6] that deciding if a digraph has a kernel is NP-complete.  ... 
arXiv:1801.02253v1 fatcat:wnu4zh5im5fw7jpnbxa65z5fwu

Page 3706 of Mathematical Reviews Vol. , Issue 99f [page]

1999 Mathematical Reviews  
More recently, Baskoro proved for d = 3 that all cycles of this automorphism (when written as a permutation of the vertex set) must have the same length.  ...  ) and every clique has a kernel.”  ... 

Quasi-Kernels for Oriented Paths and Cycles

Stephen Bowser, Charles Cable
2012 Open Journal of Discrete Mathematics  
If D is a digraph, then there is x K  such that the directed distance from y to x is less than three.  ...  We give formulae for the number of quasi-kernels and for the number of minimal quasi-kernels of oriented paths and cycles.  ...  In [3] we give sufficient conditions for a digraph to have at least three minimal quasi-kernels.  ... 
doi:10.4236/ojdm.2012.22010 fatcat:cj45gayjnrbfzh2wtwq7onxnai

Tournament limits: Degree distributions, score functions and self-converseness [article]

Erik Thörnblad
2016 arXiv   pre-print
We also show that only the uniform distribution can be realised as the outdegree distribution of a unique tournament limit.  ...  Finally we define self-converse tournament limits and kernels and characterise their degree distributions and score functions.  ...  Since t(F, G(n, W )) → t(F, W ) almost surely, we have that each tournament kernel represents some tournament limit.  ... 
arXiv:1611.09579v1 fatcat:rl3h6aswpvdqnpbpzpaltg7kqe

Polynomial kernels for deletion to classes of acyclic digraphs

Matthias Mnich, Erik Jan van Leeuwen
2017 Discrete Optimization  
We consider the problem to find a set X of vertices (or arcs) with |X| ≤ k in a given digraph G such that D = G − X is an acyclic digraph.  ...  The existence of a polynomial kernel for these problems is a notorious open problem in the field of kernelization, and little progress has been made.  ...  We consider all three problems on general digraphs, and observe that each is NP-hard, even on acyclic digraphs. More importantly, we show that all three problems admit a polynomial kernel.  ... 
doi:10.1016/j.disopt.2017.02.002 fatcat:xzrjj4b4rje55jozaq55bzth7e

Representative sets and irrelevant vertices: New tools for kernelization [article]

Stefan Kratsch, Magnus Wahlström
2012 arXiv   pre-print
We show how representative sets can be used to give a polynomial kernel for the elusive Almost 2-SAT problem.  ...  Both these kernelization results have significant spin-off effects, producing the first polynomial kernels for a range of related problems.  ...  Digraph Pair Cut has a kernel of O(k 4 ) vertices; Almost 2-SAT has a kernel of O(k 6 ) vertices.  ... 
arXiv:1111.2195v2 fatcat:ienvm4zagndaxhew246stssnka

k-kernels in k-transitive and k-quasi-transitive digraphs

César Hernández-Cruz, Hortensia Galeana-Sánchez
2012 Discrete Mathematics  
k, l)-kernel k-kernel Transitive digraph Quasi-transitive digraph a b s t r a c t Let D be a digraph, V (D) and A(D) will denote the sets of vertices and arcs of D, respectively.  ...  k ≥ 2 and that every quasi-transitive digraph has a k-kernel for every k ≥ 3.  ...  Chvátal proved in [8] that recognizing digraphs that have a kernel is an NPcomplete problem, so finding sufficient conditions for a digraph to have a kernel or finding large families of digraphs with  ... 
doi:10.1016/j.disc.2012.05.005 fatcat:b7yrcy7imbdkpnmi5rd2wtiuqa

Algorithmic aspects of quasi-kernels [article]

Hélène Langlois, Frédéric Meunier, Romeo Rizzi, Stéphane Vialette
2021 arXiv   pre-print
In 1976 Erdős and Székely conjectured that every sink-free digraph D = (V, A) has a quasi-kernel of size at most |V|/2.  ...  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.  ...  Clearly, not every digraph has a kernel (for instance, a directed cycle of odd length does not contain a kernel) and a digraph may have several kernels.  ... 
arXiv:2107.03793v1 fatcat:irb7q7xkarg4to3ieyhc2ok4ja

On the Kernel and Related Problems in Interval Digraphs [article]

Mathew C. Francis, Pavol Hell, Dalu Jacob
2021 arXiv   pre-print
of reflexive interval digraphs – which arise when we require that the two intervals assigned to a vertex have to intersect.  ...  A set S⊆ V(G) is said to be an independent set if no two vertices in S are adjacent in G. A kernel (resp. solution) of G is an independent and absorbing (resp. dominating) set in G.  ...  It is known that almost every random digraph has a kernel [17] .  ... 
arXiv:2107.08278v2 fatcat:x25a2rafafhcbnc5gtpm7rsyoy
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