4,962 research outputs found
Graph homomorphisms between trees
In this paper we study several problems concerning the number of
homomorphisms of trees. We give an algorithm for the number of homomorphisms
from a tree to any graph by the Transfer-matrix method. By using this algorithm
and some transformations on trees, we study various extremal problems about the
number of homomorphisms of trees. These applications include a far reaching
generalization of Bollob\'as and Tyomkyn's result concerning the number of
walks in trees.
Some other highlights of the paper are the following. Denote by
the number of homomorphisms from a graph to a graph . For any tree
on vertices we give a general lower bound for by certain
entropies of Markov chains defined on the graph . As a particular case, we
show that for any graph ,
where is the
largest eigenvalue of the adjacency matrix of and is a
certain constant depending only on which we call the spectral entropy of
. In the particular case when is the path on vertices, we
prove that where
is any tree on vertices, and and denote the path and star on
vertices, respectively. We also show that if is any fixed tree and
for some tree on vertices, then
must be the tree obtained from a path by attaching a pendant
vertex to the second vertex of .
All the results together enable us to show that
|\End(P_m)|\leq|\End(T_m)|\leq|\End(S_m)|, where \End(T_m) is the set of
all endomorphisms of (homomorphisms from to itself).Comment: 47 pages, 15 figure
The Complexity of Counting Homomorphisms to Cactus Graphs Modulo 2
A homomorphism from a graph G to a graph H is a function from V(G) to V(H)
that preserves edges. Many combinatorial structures that arise in mathematics
and computer science can be represented naturally as graph homomorphisms and as
weighted sums of graph homomorphisms. In this paper, we study the complexity of
counting homomorphisms modulo 2. The complexity of modular counting was
introduced by Papadimitriou and Zachos and it has been pioneered by Valiant who
famously introduced a problem for which counting modulo 7 is easy but counting
modulo 2 is intractable. Modular counting provides a rich setting in which to
study the structure of homomorphism problems. In this case, the structure of
the graph H has a big influence on the complexity of the problem. Thus, our
approach is graph-theoretic. We give a complete solution for the class of
cactus graphs, which are connected graphs in which every edge belongs to at
most one cycle. Cactus graphs arise in many applications such as the modelling
of wireless sensor networks and the comparison of genomes. We show that, for
some cactus graphs H, counting homomorphisms to H modulo 2 can be done in
polynomial time. For every other fixed cactus graph H, the problem is complete
for the complexity class parity-P which is a wide complexity class to which
every problem in the polynomial hierarchy can be reduced (using randomised
reductions). Determining which H lead to tractable problems can be done in
polynomial time. Our result builds upon the work of Faben and Jerrum, who gave
a dichotomy for the case in which H is a tree.Comment: minor change
Preservation and decomposition theorems for bounded degree structures
We provide elementary algorithms for two preservation theorems for
first-order sentences (FO) on the class \^ad of all finite structures of degree
at most d: For each FO-sentence that is preserved under extensions
(homomorphisms) on \^ad, a \^ad-equivalent existential (existential-positive)
FO-sentence can be constructed in 5-fold (4-fold) exponential time. This is
complemented by lower bounds showing that a 3-fold exponential blow-up of the
computed existential (existential-positive) sentence is unavoidable. Both
algorithms can be extended (while maintaining the upper and lower bounds on
their time complexity) to input first-order sentences with modulo m counting
quantifiers (FO+MODm). Furthermore, we show that for an input FO-formula, a
\^ad-equivalent Feferman-Vaught decomposition can be computed in 3-fold
exponential time. We also provide a matching lower bound.Comment: 42 pages and 3 figures. This is the full version of: Frederik
Harwath, Lucas Heimberg, and Nicole Schweikardt. Preservation and
decomposition theorems for bounded degree structures. In Joint Meeting of the
23rd EACSL Annual Conference on Computer Science Logic (CSL) and the 29th
Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS'14,
pages 49:1-49:10. ACM, 201
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