473 research outputs found
Complexity of planar signed graph homomorphisms to cycles
We study homomorphism problems of signed graphs. A signed graph is an
undirected graph where each edge is given a sign, positive or negative. An
important concept for signed graphs is the operation of switching at a vertex,
which is to change the sign of each incident edge. A homomorphism of a graph is
a vertex-mapping that preserves the adjacencies; in the case of signed graphs,
we also preserve the edge-signs. Special homomorphisms of signed graphs, called
s-homomorphisms, have been studied. In an s-homomorphism, we allow, before the
mapping, to perform any number of switchings on the source signed graph. This
concept has been extensively studied, and a full complexity classification
(polynomial or NP-complete) for s-homomorphism to a fixed target signed graph
has recently been obtained. Such a dichotomy is not known when we restrict the
input graph to be planar (not even for non-signed graph homomorphisms).
We show that deciding whether a (non-signed) planar graph admits a
homomorphism to the square of a cycle with , or to the circular
clique with , are NP-complete problems. We use these
results to show that deciding whether a planar signed graph admits an
s-homomorphism to an unbalanced even cycle is NP-complete. (A cycle is
unbalanced if it has an odd number of negative edges). We deduce a complete
complexity dichotomy for the planar s-homomorphism problem with any signed
cycle as a target.
We also study further restrictions involving the maximum degree and the girth
of the input signed graph. We prove that planar s-homomorphism problems to
signed cycles remain NP-complete even for inputs of maximum degree~ (except
for the case of unbalanced -cycles, for which we show this for maximum
degree~). We also show that for a given integer , the problem for signed
bipartite planar inputs of girth is either trivial or NP-complete.Comment: 17 pages, 10 figure
The chromatic polynomial of fatgraphs and its categorification
Motivated by Khovanov homology and relations between the Jones polynomial and
graph polynomials, we construct a homology theory for embedded graphs from
which the chromatic polynomial can be recovered as the Euler characteristic.
For plane graphs, we show that our chromatic homology can be recovered from the
Khovanov homology of an associated link. We apply this connection with Khovanov
homology to show that the torsion-free part of our chromatic homology is
independent of the choice of planar embedding of a graph.
We extend our construction and categorify the Bollobas-Riordan polynomial (a
generalisation of the Tutte polynomial to embedded graphs). We prove that both
our chromatic homology and the Khovanov homology of an associated link can be
recovered from this categorification.Comment: A substantial revision. To appear in Advances in Mathematic
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