596 research outputs found
Generating Functions For Kernels of Digraphs (Enumeration & Asymptotics for Nim Games)
In this article, we study directed graphs (digraphs) with a coloring
constraint due to Von Neumann and related to Nim-type games. This is equivalent
to the notion of kernels of digraphs, which appears in numerous fields of
research such as game theory, complexity theory, artificial intelligence
(default logic, argumentation in multi-agent systems), 0-1 laws in monadic
second order logic, combinatorics (perfect graphs)... Kernels of digraphs lead
to numerous difficult questions (in the sense of NP-completeness,
#P-completeness). However, we show here that it is possible to use a generating
function approach to get new informations: we use technique of symbolic and
analytic combinatorics (generating functions and their singularities) 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". Our method could be applied to more general "local coloring
constraints" in decomposable combinatorial structures.Comment: Presented (as a poster) to the conference Formal Power Series and
Algebraic Combinatorics (Vancouver, 2004), electronic proceeding
Solving the kernel perfect problem by (simple) forbidden subdigraphs for digraphs in some families of generalized tournaments and generalized bipartite tournaments
A digraph such that every proper induced subdigraph has a kernel is said to
be \emph{kernel perfect} (KP for short) (\emph{critical kernel imperfect} (CKI
for short) resp.) if the digraph has a kernel (does not have a kernel resp.).
The unique CKI-tournament is and the unique
KP-tournaments are the transitive tournaments, however bipartite tournaments
are KP. In this paper we characterize the CKI- and KP-digraphs for the
following families of digraphs: locally in-/out-semicomplete, asymmetric
arc-locally in-/out-semicomplete, asymmetric -quasi-transitive and
asymmetric -anti-quasi-transitive -free and we state that the problem
of determining whether a digraph of one of these families is CKI is polynomial,
giving a solution to a problem closely related to the following conjecture
posted by Bang-Jensen in 1998: the kernel problem is polynomially solvable for
locally in-semicomplete digraphs.Comment: 13 pages and 5 figure
k-colored kernels
We study -colored kernels in -colored digraphs. An -colored digraph
has -colored kernel if there exists a subset of its vertices such
that
(i) from every vertex there exists an at most -colored
directed path from to a vertex of and
(ii) for every there does not exist an at most -colored
directed path between them.
In this paper, we prove that for every integer there exists a -colored digraph without -colored kernel and if every directed
cycle of an -colored digraph is monochromatic, then it has a -colored
kernel for every positive integer We obtain the following results for some
generalizations of tournaments:
(i) -colored quasi-transitive and 3-quasi-transitive digraphs have a %
-colored kernel for every and respectively (we conjecture
that every -colored -quasi-transitive digraph has a % -colored kernel
for every , and
(ii) -colored locally in-tournament (out-tournament, respectively)
digraphs have a -colored kernel provided that every arc belongs to a
directed cycle and every directed cycle is at most -colored
Nondeterministic graph property testing
A property of finite graphs is called nondeterministically testable if it has
a "certificate" such that once the certificate is specified, its correctness
can be verified by random local testing. In this paper we study certificates
that consist of one or more unary and/or binary relations on the nodes, in the
case of dense graphs. Using the theory of graph limits, we prove that
nondeterministically testable properties are also deterministically testable.Comment: Version 2: 11 pages; we allow orientation in the certificate,
describe new application
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