927 research outputs found
Graph Isomorphism for Bounded Genus Graphs In Linear Time
For every integer , isomorphism of graphs of Euler genus at most can
be decided in linear time.
This improves previously known algorithms whose time complexity is
(shown in early 1980's), and in fact, this is the first fixed-parameter
tractable algorithm for the graph isomorphism problem for bounded genus graphs
in terms of the Euler genus . Our result also generalizes the seminal result
of Hopcroft and Wong in 1974, which says that the graph isomorphism problem can
be decided in linear time for planar graphs.
Our proof is quite lengthly and complicated, but if we are satisfied with an
time algorithm for the same problem, the proof is shorter and easier
MSOL Restricted Contractibility to Planar Graphs
We study the computational complexity of graph planarization via edge
contraction. The problem CONTRACT asks whether there exists a set of at
most edges that when contracted produces a planar graph. We work with a
more general problem called -RESTRICTEDCONTRACT in which , in addition,
is required to satisfy a fixed MSOL formula . We give an FPT algorithm
in time which solves -RESTRICTEDCONTRACT, where is
(i) inclusion-closed and (ii) inert contraction-closed (where inert edges are
the edges non-incident to any inclusion minimal solution ).
As a specific example, we can solve the -subgraph contractibility
problem in which the edges of a set are required to form disjoint connected
subgraphs of size at most . This problem can be solved in time using the general algorithm. We also show that for
the problem is NP-complete
Three-coloring triangle-free graphs on surfaces VII. A linear-time algorithm
We give a linear-time algorithm to decide 3-colorability of a triangle-free
graph embedded in a fixed surface, and a quadratic-time algorithm to output a
3-coloring in the affirmative case. The algorithms also allow to prescribe the
coloring of a bounded number of vertices.Comment: 22 pages, no figures; updated for reviewer remarks, reworked the
final section. arXiv admin note: text overlap with arXiv:1509.0101
Three-coloring triangle-free graphs on surfaces VI. 3-colorability of quadrangulations
We give a linear-time algorithm to decide 3-colorability (and find a
3-coloring, if it exists) of quadrangulations of a fixed surface. The algorithm
also allows to prescribe the coloring for a bounded number of vertices.Comment: 32 pages, no figures; updated for reviewer comment
The generic minimal rigidity of a partially triangulated torus
A simple graph is -rigid if its generic bar-joint frameworks in are
infinitesimally rigid. Necessary and sufficient conditions are obtained for the
minimal -rigidity of a simple graph which is obtained from the -skeleton
of a triangulated torus by the deletion of edges interior to a triangulated
disc.Comment: 31 pages, 37 figure
Spanning closed walks with bounded maximum degrees of graphs on surfaces
Gao and Richter (1994) showed that every -connected graph which embeds on
the plane or the projective plane has a spanning closed walk meeting each
vertex at most times. Brunet, Ellingham, Gao, Metzlar, and Richter (1995)
extended this result to the torus and Klein bottle. Sanders and Zhao (2001)
obtained a sharp result for higher surfaces by proving that every -connected
graph embeddable on a surface with Euler characteristic admits a
spanning closed walk meeting each vertex at most times. In this paper, we develop these results to the
remaining surfaces with Euler characteristic
Treewidth of grid subsets
Let Q_n be the graph of n times n times n cube with all non-decreasing
diagonals (including the facial ones) in its constituent unit cubes. Suppose
that a subset S of V(Q_n) separates the left side of the cube from the right
side. We show that S induces a subgraph of tree-width at least n/sqrt{18}-1. We
use a generalization of this claim to prove that the vertex set of Q_n cannot
be partitioned to two parts, each of them inducing a subgraph of bounded
tree-width.Comment: 15 pages, no figure
Nerves, minors, and piercing numbers
We make the first step towards a "nerve theorem" for graphs. Let be a
simple graph and let be a family of induced subgraphs of such
that the intersection of any members of is either empty or
connected. We show that if the nerve complex of has non-vanishing
homology in dimension three, then contains the complete graph on five
vertices as a minor. As a consequence we confirm a conjecture of Goaoc
concerning an extension of the planar theorem due to Alon and Kleitman:
Let be a finite family of open connected sets in the plane such
that the intersection of any members of is either empty or
connected. If among any members of there are some
three that intersect, then there is a set of points which intersects every
member of , where is a constant depending only on
The Kuenneth formula for graphs
We construct a Cartesian product G x H for finite simple graphs. It satisfies
the Kuenneth formula: H^k(G x H) is a direct sum of tensor products H^i(G) x
H^j(G) with i+j=k and so p(G x H,x) = p(G,x) p(H,y) for the Poincare polynomial
p(G,x) and X(G x H) = X(G) X(H) for the Euler characteristic X(G)=p(G,-1). G1=G
x K1 has as vertices the simplices of G and a natural digraph structure. We
show that dim(G1) is larger or equal than dim(G) and G1 is homotopic to G. The
Kuenneth identity is proven using Hodge describing the harmonic forms by the
product f g of harmonic forms of G and H and uses a discrete de Rham theorem
given by a combinatorial chain homotopy between simplicial and de Rham
cohomology. We show dim(G x H) = dim(G1) + dim(H1) implying that dim(G x H) is
larger or equal than dim(G) + dim(H) as for Hausdorff dimension in the
continuum. The chromatic number c(G1) is smaller or equal than c(G) and c(G x
H) is bounded above by c(G)+c(H)-1. The automorphism group of G x H contains
Aut(G) x Aut(H). If G~H and U~V then (G x U) ~ (H x V) if ~ means homotopic:
homotopy classes can be multiplided. If G is k-dimensional geometric meaning
that all unit spheres S(x) in G are (k-1)-discrete homotopy spheres, then G1 is
k-dimensional geometric. If G is k-dimensional geometric and H is l-dimensional
geometric, then G x H is geometric of dimension (l+k). The product extends to a
ring of chains which unlike the category of graphs is closed under boundary
operation taking quotients G/A with A subset Aut(G). As we can glue graphs or
chains, joins or fibre bundles can be defined with the same features as in the
continuum, allowing to build isomorphism classes of bundles.Comment: 60 pages 56 figure
Coloring graphs using topology
Higher dimensional graphs can be used to colour two-dimensional geometric
graphs. If G the boundary of a three dimensional graph H for example, we can
refine the interior until it is colourable with 4 colours. The later goal is
achieved if all interior edge degrees are even. Using a refinement process
which cuts the interior along surfaces we can adapt the degrees along the
boundary of that surface. More efficient is a self-cobordism of G with itself
with a host graph discretizing the product of G with an interval. It follows
from the fact that Euler curvature is zero everywhere for three dimensional
geometric graphs, that the odd degree edge set O is a cycle and so a boundary
if H is simply connected. A reduction to minimal colouring would imply the four
colour theorem. The method is expected to give a reason "why 4 colours suffice"
and suggests that every two dimensional geometric graph of arbitrary degree and
orientation can be coloured by 5 colours: since the projective plane can not be
a boundary of a 3-dimensional graph and because for higher genus surfaces, the
interior H is not simply connected, we need in general to embed a surface into
a 4-dimensional simply connected graph in order to colour it. This explains the
appearance of the chromatic number 5 for higher degree or non-orientable
situations, a number we believe to be the upper limit. For every surface type,
we construct examples with chromatic number 3,4 or 5, where the construction of
surfaces with chromatic number 5 is based on a method of Fisk. We have
implemented and illustrated all the topological aspects described in this paper
on a computer. So far we still need human guidance or simulated annealing to do
the refinements in the higher dimensional host graph.Comment: 81 pages, 48 figure
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