152 research outputs found
Arc-Disjoint Paths and Trees in 2-Regular Digraphs
An out-(in-)branching B_s^+ (B_s^-) rooted at s in a digraph D is a connected
spanning subdigraph of D in which every vertex x != s has precisely one arc
entering (leaving) it and s has no arcs entering (leaving) it. We settle the
complexity of the following two problems:
1) Given a 2-regular digraph , decide if it contains two arc-disjoint
branchings B^+_u, B^-_v.
2) Given a 2-regular digraph D, decide if it contains an out-branching B^+_u
such that D remains connected after removing the arcs of B^+_u.
Both problems are NP-complete for general digraphs. We prove that the first
problem remains NP-complete for 2-regular digraphs, whereas the second problem
turns out to be polynomial when we do not prescribe the root in advance. We
also prove that, for 2-regular digraphs, the latter problem is in fact
equivalent to deciding if contains two arc-disjoint out-branchings. We
generalize this result to k-regular digraphs where we want to find a number of
pairwise arc-disjoint spanning trees and out-branchings such that there are k
in total, again without prescribing any roots.Comment: 9 pages, 7 figure
On Complexity of Minimum Leaf Out-branching Problem
Given a digraph , the Minimum Leaf Out-Branching problem (MinLOB) is the
problem of finding in an out-branching with the minimum possible number of
leaves, i.e., vertices of out-degree 0. Gutin, Razgon and Kim (2008) proved
that MinLOB is polynomial time solvable for acyclic digraphs which are exactly
the digraphs of directed path-width (DAG-width, directed tree-width,
respectively) 0. We investigate how much one can extend this polynomiality
result. We prove that already for digraphs of directed path-width (directed
tree-width, DAG-width, respectively) 1, MinLOB is NP-hard. On the other hand,
we show that for digraphs of restricted directed tree-width (directed
path-width, DAG-width, respectively) and a fixed integer , the problem of
checking whether there is an out-branching with at most leaves is
polynomial time solvable
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 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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