41 research outputs found
A note on the flip distance between non-crossing spanning trees
We consider spanning trees of points in convex position whose edges are
pairwise non-crossing. Applying a flip to such a tree consists in adding an
edge and removing another so that the result is still a non-crossing spanning
tree. Given two trees, we investigate the minimum number of flips required to
transform one into the other. The naive upper bound stood for 25
years until a recent breakthrough from Aichholzer et al. yielding a
bound. We improve their result with a
upper bound, and we strengthen and shorten the proofs of several of their
results
The smallest 5-chromatic tournament
A coloring of a digraph is a partition of its vertex set such that each class
induces a digraph with no directed cycles. A digraph is -chromatic if is
the minimum number of classes in such partition, and a digraph is oriented if
there is at most one arc between each pair of vertices. Clearly, the smallest
-chromatic digraph is the complete digraph on vertices, but determining
the order of the smallest -chromatic oriented graphs is a challenging
problem. It is known that the smallest -, - and -chromatic oriented
graphs have , and vertices, respectively. In 1994, Neumann-Lara
conjectured that a smallest -chromatic oriented graph has vertices. We
solve this conjecture and show that the correct order is
Reconfiguration of plane trees in convex geometric graphs
A non-crossing spanning tree of a set of points in the plane is a spanning
tree whose edges pairwise do not cross. Avis and Fukuda in 1996 proved that
there always exists a flip sequence of length at most between any pair
of non-crossing spanning trees (where denotes the number of points).
Hernando et al. proved that the length of a minimal flip sequence can be of
length at least . Two recent results of Aichholzer et al. and
Bousquet et al. improved the Avis and Fukuda upper bound by proving that there
always exists a flip sequence of length respectively at most and
. We improve the upper bound by a linear factor for the first
time in 25 years by proving that there always exists a flip sequence between
any pair of non-crossing spanning trees of length at most where
. Our result is actually stronger since we prove that, for any
two trees , there exists a flip sequence from to of length
at most . We also improve the best lower bound in terms
of the symmetric difference by proving that there exists a pair of trees
such that a minimal flip sequence has length , improving the lower bound of Hernando et al. by considering the
symmetric difference instead of the number of vertices. We generalize this
lower bound construction to non-crossing flips (where we close the gap between
upper and lower bounds) and rotations
Parameterized complexity of edge-coloured and signed graph homomorphism problems
We study the complexity of graph modification problems for homomorphism-based
properties of edge-coloured graphs. A homomorphism from an edge-coloured graph
to an edge-coloured graph is a vertex-mapping from to that
preserves adjacencies and edge-colours. We consider the property of having a
homomorphism to a fixed edge-coloured graph . Given an edge-coloured graph
, can we perform graph operations so that the resulting graph has a
homomorphism to ? The operations we consider are vertex-deletion,
edge-deletion and switching (an operation that permutes the colours of the
edges incident to a given vertex). Switching plays an important role in the
theory of signed graphs, that are -edge-coloured graphs whose colours are
and . We denote the corresponding problems (parameterized by ) by
VERTEX DELETION -COLOURING, EDGE DELETION -COLOURING and SWITCHING
-COLOURING. These generalise -COLOURING (where one has to decide if an
input graph admits a homomorphism to ). Our main focus is when has order
at most , a case that includes standard problems such as VERTEX COVER, ODD
CYCLE TRANSVERSAL and EDGE BIPARTIZATION. For such a graph , we give a
P/NP-complete complexity dichotomy for all three studied problems. Then, we
address their parameterized complexity. We show that all VERTEX DELETION
-COLOURING and EDGE DELETION -COLOURING problems for such are FPT.
This is in contrast with the fact that already for some of order~,
unless P=NP, none of the three considered problems is in XP. We show that the
situation is different for SWITCHING -COLOURING: there are three
-edge-coloured graphs of order for which this is W-hard, and
assuming the ETH, admits no algorithm in time for inputs of size
. For the other cases, SWITCHING -COLOURING is FPT.Comment: 18 pages, 8 figures, 1 table. To appear in proceedings of IPEC 201
Graph modification for edge-coloured and signed graph homomorphism problems: parameterized and classical complexity
We study the complexity of graph modification problems with respect to
homomorphism-based colouring properties of edge-coloured graphs. A homomorphism
from edge-coloured graph to edge-coloured graph is a vertex-mapping
from to that preserves adjacencies and edge-colours. We consider the
property of having a homomorphism to a fixed edge-coloured graph . The
question we are interested in is: given an edge-coloured graph , can we
perform graph operations so that the resulting graph admits a homomorphism
to ? The operations we consider are vertex-deletion, edge-deletion and
switching (an operation that permutes the colours of the edges incident to a
given vertex). Switching plays an important role in the theory of signed
graphs, that are 2-edge-coloured graphs whose colours are the signs and
. We denote the corresponding problems (parameterized by ) by
VD--COLOURING, ED--COLOURING and SW--COLOURING. These problems
generalise -COLOURING (to decide if an input graph admits a homomorphism to
a fixed target ).
Our main focus is when is an edge-coloured graph with at most two
vertices, a case that is already interesting as it includes problems such as
VERTEX COVER, ODD CYCLE RANSVERSAL and EDGE BIPARTIZATION. For such a graph
, we give a P/NP-c complexity dichotomy for VD--COLOURING,
ED--COLOURING and SW--COLOURING. We then address their parameterized
complexity. We show that VD--COLOURING and ED--COLOURING for all such
are FPT. In contrast, already for some of order 3, unless P=NP, none of the
three problems is in XP, since 3-COLOURING is NP-c. We show that
SW--COLOURING is different: there are three 2-edge-coloured graphs of
order 2 for which SW--COLOURING is W-hard, and assuming the ETH, admits no
algorithm in time . For the other cases, SW--COLOURING is FPT.Comment: 17 pages, 9 figures, 2 table