120 research outputs found
On the Complexity of Digraph Colourings and Vertex Arboricity
It has been shown by Bokal et al. that deciding 2-colourability of digraphs
is an NP-complete problem. This result was later on extended by Feder et al. to
prove that deciding whether a digraph has a circular -colouring is
NP-complete for all rational . In this paper, we consider the complexity
of corresponding decision problems for related notions of fractional colourings
for digraphs and graphs, including the star dichromatic number, the fractional
dichromatic number and the circular vertex arboricity. We prove the following
results:
Deciding if the star dichromatic number of a digraph is at most is
NP-complete for every rational .
Deciding if the fractional dichromatic number of a digraph is at most is
NP-complete for every .
Deciding if the circular vertex arboricity of a graph is at most is
NP-complete for every rational .
To show these results, different techniques are required in each case. In
order to prove the first result, we relate the star dichromatic number to a new
notion of homomorphisms between digraphs, called circular homomorphisms, which
might be of independent interest. We provide a classification of the
computational complexities of the corresponding homomorphism colouring problems
similar to the one derived by Feder et al. for acyclic homomorphisms.Comment: 21 pages, 1 figur
Optimal path and cycle decompositions of dense quasirandom graphs
Motivated by longstanding conjectures regarding decompositions of graphs into
paths and cycles, we prove the following optimal decomposition results for
random graphs. Let be constant and let . Let be
the number of odd degree vertices in . Then a.a.s. the following hold:
(i) can be decomposed into cycles and a
matching of size .
(ii) can be decomposed into
paths.
(iii) can be decomposed into linear forests.
Each of these bounds is best possible. We actually derive (i)--(iii) from
`quasirandom' versions of our results. In that context, we also determine the
edge chromatic number of a given dense quasirandom graph of even order. For all
these results, our main tool is a result on Hamilton decompositions of robust
expanders by K\"uhn and Osthus.Comment: Some typos from the first version have been correcte
Disimplicial arcs, transitive vertices, and disimplicial eliminations
In this article we deal with the problems of finding the disimplicial arcs of
a digraph and recognizing some interesting graph classes defined by their
existence. A diclique of a digraph is a pair of sets of vertices such
that is an arc for every and . An arc is
disimplicial when is a diclique. We show that the problem
of finding the disimplicial arcs is equivalent, in terms of time and space
complexity, to that of locating the transitive vertices. As a result, an
efficient algorithm to find the bisimplicial edges of bipartite graphs is
obtained. Then, we develop simple algorithms to build disimplicial elimination
schemes, which can be used to generate bisimplicial elimination schemes for
bipartite graphs. Finally, we study two classes related to perfect disimplicial
elimination digraphs, namely weakly diclique irreducible digraphs and diclique
irreducible digraphs. The former class is associated to finite posets, while
the latter corresponds to dedekind complete finite posets.Comment: 17 pags., 3 fig
On Spanning Galaxies in Digraphs
International audienceIn a directed graph, a star is an arborescence with at least one arc, in which the root dominates all the other vertices. A galaxy is a vertex-disjoint union of stars. In this paper, we consider the Spanning Galaxy problem of deciding whether a digraph D has a spanning galaxy or not. We show that although this problem is NP-complete (even when restricted to acyclic digraphs), it becomes polynomial-time solvable when restricted to strong digraphs. In fact, we prove that restricted to this class, the \pb\ is equivalent to the problem of deciding if a strong digraph has a strong digraph with an even number of vertices. We then show a polynomial-time algorithm to solve this problem. We also consider some parameterized version of the Spanning Galaxy problem. Finally, we improve some results concerning the notion of directed star arboricity of a digraph D, which is the minimum number of galaxies needed to cover all the arcs of D. We show in particular that dst(D)\leq \Delta(D)+1 for every digraph D and that dst(D)\leq\Delta(D) for every acyclic digraph D
The degree-diameter problem for sparse graph classes
The degree-diameter problem asks for the maximum number of vertices in a
graph with maximum degree and diameter . For fixed , the answer
is . We consider the degree-diameter problem for particular
classes of sparse graphs, and establish the following results. For graphs of
bounded average degree the answer is , and for graphs of
bounded arboricity the answer is \Theta(\Delta^{\floor{k/2}}), in both cases
for fixed . For graphs of given treewidth, we determine the the maximum
number of vertices up to a constant factor. More precise bounds are given for
graphs of given treewidth, graphs embeddable on a given surface, and
apex-minor-free graphs
Interpolation theorem for a continuous function on orientations of a simple graph
summary:Let be a simple graph. A function from the set of orientations of to the set of non-negative integers is called a continuous function on orientations of if, for any two orientations and of , whenever and differ in the orientation of exactly one edge of . We show that any continuous function on orientations of a simple graph has the interpolation property as follows: If there are two orientations and of with and , where , then for any integer such that , there are at least orientations of satisfying , where equals the number of edges of . It follows that some useful invariants of digraphs including the connectivity, the arc-connectivity and the absorption number, etc., have the above interpolation property on the set of all orientations of
Graph Treewidth and Geometric Thickness Parameters
Consider a drawing of a graph in the plane such that crossing edges are
coloured differently. The minimum number of colours, taken over all drawings of
, is the classical graph parameter "thickness". By restricting the edges to
be straight, we obtain the "geometric thickness". By further restricting the
vertices to be in convex position, we obtain the "book thickness". This paper
studies the relationship between these parameters and treewidth.
Our first main result states that for graphs of treewidth , the maximum
thickness and the maximum geometric thickness both equal .
This says that the lower bound for thickness can be matched by an upper bound,
even in the more restrictive geometric setting. Our second main result states
that for graphs of treewidth , the maximum book thickness equals if and equals if . This refutes a conjecture of Ganley and
Heath [Discrete Appl. Math. 109(3):215-221, 2001]. Analogous results are proved
for outerthickness, arboricity, and star-arboricity.Comment: A preliminary version of this paper appeared in the "Proceedings of
the 13th International Symposium on Graph Drawing" (GD '05), Lecture Notes in
Computer Science 3843:129-140, Springer, 2006. The full version was published
in Discrete & Computational Geometry 37(4):641-670, 2007. That version
contained a false conjecture, which is corrected on page 26 of this versio
An Efficient Algorithm for 1-Dimensional (Persistent) Path Homology
This paper focuses on developing an efficient algorithm for analyzing a
directed network (graph) from a topological viewpoint. A prevalent technique
for such topological analysis involves computation of homology groups and their
persistence. These concepts are well suited for spaces that are not directed.
As a result, one needs a concept of homology that accommodates orientations in
input space. Path-homology developed for directed graphs by Grigor'yan, Lin,
Muranov and Yau has been effectively adapted for this purpose recently by
Chowdhury and M\'emoli. They also give an algorithm to compute this
path-homology. Our main contribution in this paper is an algorithm that
computes this path-homology and its persistence more efficiently for the
-dimensional () case. In developing such an algorithm, we discover
various structures and their efficient computations that aid computing the
-dimensional path-homnology. We implement our algorithm and present some
preliminary experimental results
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