23 research outputs found
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
Arc-disjoint out- and in-branchings in compositions of digraphs
An out-branching (in-branching ) in a digraph is a
connected spanning subdigraph of in which every vertex except the vertex
, called the root, has in-degree (out-degree) one. A {\bf good
-pair} in is a pair of branchings which have
no arc in common.
Thomassen proved that is NP-complete to decide if a digraph has any good
pair. A digraph is {\bf semicomplete} if it has no pair of non adjacent
vertices. A {\bf semicomplete composition} is any digraph which is obtained
from a semicomplete digraph by substituting an arbitrary digraph for
each vertex of .
Recently the authors of this paper gave a complete classification of
semicomplete digraphs which have a good -pair, where are
prescribed vertices of . They also gave a polynomial algorithm which for a
given semicomplete digraph and vertices of , either produces a
good -pair in or a certificate that has such pair. In this paper
we show how to use the result for semicomplete digraphs to completely solve the
problem of deciding whether a given semicomplete composition , has a good
-pair for given vertices of . Our solution implies that the
problem is polynomially solvable for all semicomplete compositions. In
particular our result implies that there is a polynomial algorithm for deciding
whether a given quasi-transitive digraph has a good -pair for given
vertices of . This confirms a conjecture of Bang-Jensen and Gutin from
1998
Arc-disjoint Strong Spanning Subdigraphs of Semicomplete Compositions
A strong arc decomposition of a digraph is a decomposition of its
arc set into two disjoint subsets and such that both of the
spanning subdigraphs and are strong. 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 We
obtain a characterization of digraph compositions which
have a strong arc decomposition when is a semicomplete digraph and each
is an arbitrary digraph. Our characterization generalizes a
characterization by Bang-Jensen and Yeo (2003) of semicomplete digraphs with a
strong arc decomposition and solves an open problem by Sun, Gutin and Ai (2018)
on strong arc decompositions of digraph compositions in
which is semicomplete and each is arbitrary. Our proofs are
constructive and imply the existence of a polynomial algorithm for constructing
a \good{} decomposition of a digraph , with
semicomplete, whenever such a decomposition exists
Strong arc decompositions of split digraphs
A {\bf strong arc decomposition} of a digraph is a partition of its
arc set into two sets such that the digraph is
strong for . Bang-Jensen and Yeo (2004) conjectured that there is some
such that every -arc-strong digraph has a strong arc decomposition. They
also proved that with one exception on 4 vertices every 2-arc-strong
semicomplete digraph has a strong arc decomposition. Bang-Jensen and Huang
(2010) extended this result to locally semicomplete digraphs by proving that
every 2-arc-strong locally semicomplete digraph which is not the square of an
even cycle has a strong arc decomposition. This implies that every 3-arc-strong
locally semicomplete digraph has a strong arc decomposition. A {\bf split
digraph} is a digraph whose underlying undirected graph is a split graph,
meaning that its vertices can be partioned into a clique and an independent
set. Equivalently, a split digraph is any digraph which can be obtained from a
semicomplete digraph by adding a new set of vertices and some
arcs between and . In this paper we prove that every 3-arc-strong split
digraph has a strong arc decomposition which can be found in polynomial time
and we provide infinite classes of 2-strong split digraphs with no strong arc
decomposition. We also pose a number of open problems on split digraphs
Arc-disjoint strong spanning subdigraphs in compositions and products of digraphs
A digraph has a good decomposition if has two disjoint sets
and such that both and are strong. 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
For digraph compositions , we obtain sufficient
conditions for to have a good decomposition and a characterization of
with a good decomposition when is a strong semicomplete digraph and each
is an arbitrary digraph with at least two vertices.
For digraph products, we prove the following: (a) if is an integer
and is a strong digraph which has a collection of arc-disjoint cycles
covering all vertices, then the Cartesian product digraph (the
th powers with respect to Cartesian product) has a good decomposition; (b)
for any strong digraphs , the strong product has a good
decomposition