16 research outputs found
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
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
Out-degree reducing partitions of digraphs
Let be a fixed integer. We determine the complexity of finding a
-partition of the vertex set of a given digraph such
that the maximum out-degree of each of the digraphs induced by , () is at least smaller than the maximum out-degree of . We show
that this problem is polynomial-time solvable when and -complete otherwise. The result for and answers a question
posed in \cite{bangTCS636}. We also determine, for all fixed non-negative
integers , the complexity of deciding whether a given digraph of
maximum out-degree has a -partition such that the digraph
induced by has maximum out-degree at most for . It
follows from this characterization that the problem of deciding whether a
digraph has a 2-partition such that each vertex has at
least as many neighbours in the set as in , for is
-complete. This solves a problem from \cite{kreutzerEJC24} on
majority colourings.Comment: 11 pages, 1 figur
The complexity of finding arc-disjoint branching flows
The concept of arc-disjoint flows in networks was recently introduced in \cite{bangTCSflow}. This is a very general framework within which many well-known and important problems can be formulated. In particular, the existence of arc-disjoint branching flows, that is, flows which send one unit of flow from a given source to all other vertices, generalizes the concept of arc-disjoint out-branchings (spanning out-trees) in a digraph. A pair of out-branchings from a root in a digraph on vertices corresponds to arc-disjoint branching flows (the arcs carrying flow in are those used in , ) in the network that we obtain from by giving all arcs capacity .It is then a natural question to ask how much we can lower the capacities on the arcs and still have, say, two arc-disjoint branching flows from the given root .We prove that for every fixed integer it is\begin{itemize}\item an NP-complete problem to decide whether a network where for every arc has two arc-disjoint branching flows rooted at .\item a polynomial problem to decide whether a network on vertices and for every arc has two arc-disjoint branching flows rooted at .\end{itemize}The algorithm for the later result generalizes the polynomial algorithm, due to Lov\'asz, for deciding whether a given input digraph has two arc-disjoint out-branchings rooted at a given vertex.Finally we prove that under the so-called Exponential Time Hypothesis (ETH), for every and for every with (and for every large we have for some ) there is no polynomial algorithm for deciding whether a given digraph contains twoarc-disjoint branching flows from the same root so that no arc carries flow larger than
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
Tournaments and Semicomplete Digraphs.
International audienc