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Almost all digraphs have a kernel

1990
*
Discrete Mathematics
*

In this paper it is shown that

doi:10.1016/0012-365x(90)90373-p
fatcat:xjjhuzqqkvayxo622oozpuohwe
*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. ...##
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Generating Functions For Kernels of Digraphs (Enumeration & Asymptotics for Nim Games)
[article]

2004
*
arXiv
*
pre-print

) in order to get exact and asymptotic results, e.g. for the existence of

arXiv:math/0411138v1
fatcat:z6qkbcfuwjfdpaz2hlmxixj2xa
*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. ...##
###
Independent and monochromatic absorbent sets in infinite digraphs

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 ...

##
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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

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] . ...

##
###
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. ...

##
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Perfect graphs with polynomially computable kernels
[article]

2018
*
arXiv
*
pre-print

Not

arXiv:1801.02253v1
fatcat:wnu4zh5im5fw7jpnbxa65z5fwu
*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. ...##
###
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

2012
*
Open Journal of Discrete Mathematics
*

If D is

doi:10.4236/ojdm.2012.22010
fatcat:cj45gayjnrbfzh2wtwq7onxnai
*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*. ...##
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Tournament limits: Degree distributions, score functions and self-converseness
[article]

2016
*
arXiv
*
pre-print

We also show that only the uniform distribution can be realised as the outdegree distribution of

arXiv:1611.09579v1
fatcat:rl3h6aswpvdqnpbpzpaltg7kqe
*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. ...##
###
Polynomial kernels for deletion to classes of acyclic digraphs

2017
*
Discrete Optimization
*

We consider the problem to find

doi:10.1016/j.disopt.2017.02.002
fatcat:xzrjj4b4rje55jozaq55bzth7e
*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*. ...##
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Representative sets and irrelevant vertices: New tools for kernelization
[article]

2012
*
arXiv
*
pre-print

We show how representative sets can be used to give

arXiv:1111.2195v2
fatcat:ienvm4zagndaxhew246stssnka
*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. ...##
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k-kernels in k-transitive and k-quasi-transitive digraphs

2012
*
Discrete Mathematics
*

k, l)-

doi:10.1016/j.disc.2012.05.005
fatcat:b7yrcy7imbdkpnmi5rd2wtiuqa
*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 ...##
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Algorithmic aspects of quasi-kernels
[article]

2021
*
arXiv
*
pre-print

In 1976 Erdős and Székely conjectured that every sink-free

arXiv:2107.03793v1
fatcat:irb7q7xkarg4to3ieyhc2ok4ja
*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*. ...##
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On the Kernel and Related Problems in Interval Digraphs
[article]

2021
*
arXiv
*
pre-print

of reflexive interval

arXiv:2107.08278v2
fatcat:x25a2rafafhcbnc5gtpm7rsyoy
*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] . ...
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