12 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
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
Edge disjoint In- and Out-branchings in Tournaments and related path problems
AbstractWe give good characterizations in Edmonds' sense for the following problems for tournaments. 1.(1) Given a tournament T and v1, v2 ∈ V(T), possibly v1 = v2, when do there exist edge-disjoint branchings Fv1+, Fv2−, such that Fv1+ is an out-branching rooted at v1 and Fv2− is an in-branching rooted at v2?2.(2) Given a strong tournament T and x1, x2, y1, y2 ∈ V(T) all different, when do there exist edge-disjoint (x1, y1)-, (x2, y2)-paths in T?3.(3) Given a strong tournament T and a, b, c ∈ V(T) all different, when do there exist edge-disjoint (a, b)-, (b, c)-paths in T?Problem (2) is known to be NP-complete for digraphs in general and we give a proof by C. Thomassen that problem (1) is NP-complete for digraphs in general