77 research outputs found
On the existence and number of -kings in -quasi-transitive digraphs
Let be a digraph and an integer. We say that
is -quasi-transitive if for every directed path in
, then or . Clearly, a
2-quasi-transitive digraph is a quasi-transitive digraph in the usual sense.
Bang-Jensen and Gutin proved that a quasi-transitive digraph has a 3-king
if and only if has a unique initial strong component and, if has a
3-king and the unique initial strong component of has at least three
vertices, then has at least three 3-kings. In this paper we prove the
following generalization: A -quasi-transitive digraph has a -king
if and only if has a unique initial strong component, and if has a
-king then, either all the vertices of the unique initial strong
components are -kings or the number of -kings in is at least
.Comment: 17 page
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
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
Minimum Cost Homomorphisms to Locally Semicomplete and Quasi-Transitive Digraphs
For digraphs and , a homomorphism of to is a mapping $f:\
V(G)\dom V(H)uv\in A(G)f(u)f(v)\in A(H)u \in V(G)c_i(u), i \in V(H)f\sum_{u\in V(G)}c_{f(u)}(u)HHHGc_i(u)u\in V(G)i\in V(H)GH$ and, if one exists, to find one of minimum cost.
Minimum cost homomorphism problems encompass (or are related to) many well
studied optimization problems such as the minimum cost chromatic partition and
repair analysis problems. We focus on the minimum cost homomorphism problem for
locally semicomplete digraphs and quasi-transitive digraphs which are two
well-known generalizations of tournaments. Using graph-theoretic
characterization results for the two digraph classes, we obtain a full
dichotomy classification of the complexity of minimum cost homomorphism
problems for both classes
Arc-disjoint in- and out-branchings rooted at the same vertex in compositions of digraphs
A digraph has a good pair at a vertex if has a pair of
arc-disjoint in- and out-branchings rooted at . Let be a digraph with
vertices and let be digraphs such that
has vertices Then the composition
is a digraph with vertex set and arc set
When is arbitrary, we obtain the following result: every strong digraph
composition in which for every , has a good pair
at every vertex of The condition of in this result cannot be
relaxed. When is semicomplete, we characterize semicomplete compositions
with a good pair, which generalizes the corresponding characterization by
Bang-Jensen and Huang (J. Graph Theory, 1995) for quasi-transitive digraphs. As
a result, we can decide in polynomial time whether a given semicomplete
composition has a good pair rooted at a given vertex
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