54 research outputs found
On retracts, absolute retracts, and folds in cographs
Let G and H be two cographs. We show that the problem to determine whether H
is a retract of G is NP-complete. We show that this problem is fixed-parameter
tractable when parameterized by the size of H. When restricted to the class of
threshold graphs or to the class of trivially perfect graphs, the problem
becomes tractable in polynomial time. The problem is also soluble when one
cograph is given as an induced subgraph of the other. We characterize absolute
retracts of cographs.Comment: 15 page
Beyond Helly graphs: the diameter problem on absolute retracts
Characterizing the graph classes such that, on -vertex -edge graphs in
the class, we can compute the diameter faster than in time is an
important research problem both in theory and in practice. We here make a new
step in this direction, for some metrically defined graph classes.
Specifically, a subgraph of a graph is called a retract of if it is
the image of some idempotent endomorphism of . Two necessary conditions for
being a retract of is to have is an isometric and isochromatic
subgraph of . We say that is an absolute retract of some graph class
if it is a retract of any of which it is an
isochromatic and isometric subgraph. In this paper, we study the complexity of
computing the diameter within the absolute retracts of various hereditary graph
classes. First, we show how to compute the diameter within absolute retracts of
bipartite graphs in randomized time. For the
special case of chordal bipartite graphs, it can be improved to linear time,
and the algorithm even computes all the eccentricities. Then, we generalize
these results to the absolute retracts of -chromatic graphs, for every fixed
. Finally, we study the diameter problem within the absolute retracts
of planar graphs and split graphs, respectively
-Category of Moment-Angle Manifolds, Massey Products, and a Generalization of the Golod Property
We give various bounds for the Lusternik-Schnirelmann category of
moment-angle complexes and show how this relates to vanishing of Massey
products in . In particular, we
characterise the Lusternik-Schnirelmann category of moment-angle manifolds
over triangulated -spheres for , as well as
higher dimension spheres built up via connected sum, join, and vertex doubling
operations. This characterisation is given in terms of the combinatorics of
, the cup product length of , as well as a certain
generalisation of the Golod property. Some applications include information
about the category and vanishing of Massey products for moment-angle complexes
over fullerenes and -neighbourly complexes.Comment: New examples adde
Building blocks for the variety of absolute retracts
AbstractGiven a graph H with a labelled subgraph G, a retraction of H to G is a homomorphism r:H→G such that r(x)=x for all vertices x in G. We call G a retract of H. While deciding the existence of a retraction to a fixed graph G is NP-complete in general, necessary and sufficient conditions have been provided for certain classes of graphs in terms of holes, see for example Hell and Rival.For any integer k⩾2 we describe a collection of graphs that generate the variety ARk of graphs G with the property that G is a retract of H whenever G is a subgraph of H and no hole in G of size at most k is filled by a vertex of H. We also prove that ARk⊂NUFk+1, where NUFk+1 is the variety of graphs that admit a near unanimity function of arity k+1
Loop space decompositions of moment-angle complexes associated to graphs
We prove that the loop space of moment-angle complexes associated to a graph
is homotopy equivalent to a finite type product of spheres and loops on simply
connected spheres. To do this, a general result showing that the retract of a
finite type product of spheres and loops on simply connected spheres is of the
same form is proved.Comment: 21 pages, comments welcom
The homotopy theory of polyhedral products associated with flag complexes
If is a simplicial complex on vertices the flagification of is
the minimal flag complex on the same vertex set that contains .
Letting be the set of vertices, there is a sequence of simplicial
inclusions . This induces a sequence of maps of polyhedral
products .
We show that and have right homotopy
inverses and draw consequences. For a flag complex the polyhedral product
of the form is a co--space if and only if
the -skeleton of is a chordal graph, and we deduce that the maps and
have right homotopy inverses in this case.Comment: 25 page
A Brightwell-Winkler type characterisation of NU graphs
In 2000, Brightwell and Winkler characterised dismantlable graphs as the
graphs for which the Hom-graph , defined on the set of
homomorphisms from to , is connected for all graphs . This shows that
the reconfiguration version of the -colouring
problem, in which one must decide for a given whether is
connected, is trivial if and only if is dismantlable.
We prove a similar starting point for the reconfiguration version of the
-extension problem. Where is the subgraph of the
Hom-graph induced by the -colourings extending the
-precolouring of , the reconfiguration version
of the -extension problem asks, for a given -precolouring of a graph
, if is connected. We show that the graphs for which
is connected for every choice of are exactly the
graphs. This gives a new characterisation of graphs, a
nice class of graphs that is important in the algebraic approach to the -dichotomy.
We further give bounds on the diameter of for
graphs , and show that shortest path between two vertices of can be found in parameterised polynomial time. We apply our
results to the problem of shortest path reconfiguration, significantly
extending recent results.Comment: 17 pages, 1 figur
LS-category of moment-angle manifolds, Massey products, and a generalisation of the Golod property
This paper is obtained as as synergy of homotopy theory, commutative algebra and combinatorics. We give various bounds for the Lusternik-Schnirelmann category of moment-angle complexes and show how this relates to vanishing of Massey products in Tor+R[v1,...,vn] (R [K], R) for the Stanley-Reisner ring R[K]. In particular, we characterise the Lusternik-Schnirelmann category of moment-angle manifolds ZK over triangulated d-spheres K for d ≤ 2, as well as higher dimension spheres built up via connected sum, join, and vertex doubling operations. This characterisation is given in terms of the combinatorics of K, the cup product length of H* (ZK), as well as a certain generalisation of the Golod property. As an application, we describe conditions for vanishing of Massey products in the case of fullerenes and k-neighbourly complexes
- …