56,847 research outputs found
Properties of graphs with large girth
This thesis is devoted to the analysis of a class of
iterative probabilistic algorithms in regular graphs, called
locally greedy algorithms, which will provide bounds for
graph functions in regular graphs with large girth. This class is
useful because, by conveniently setting the parameters associated
with it, we may derive algorithms for some well-known graph
problems, such as algorithms to find a large independent set, a
large induced forest, or even a small dominating set in an input
graph G. The name ``locally greedy" comes from the fact that, in
an algorithm of this class, the probability associated with the
random selection of a vertex v is determined by the current
state of the vertices within some fixed distance of v.
Given r > 2 and an r-regular graph G, we determine the
expected performance of a locally greedy algorithm in G,
depending on the girth g of the input and on the degree r of
its vertices. When the girth of the graph is sufficiently large,
this analysis leads to new lower bounds on the independence number
of G and on the maximum number of vertices in an induced forest
in G, which, in both cases, improve the bounds previously known.
It also implies bounds on the same functions in graphs with large
girth and maximum degree r and in random regular graphs. As a
matter of fact, the asymptotic lower bounds on the cardinality of
a maximum induced forest in a random regular graph improve earlier
bounds, while, for independent sets, our bounds coincide with
asymptotic lower bounds first obtained by Wormald. Our result
provides an alternative proof of these bounds which avoids sharp
concentration arguments.
The main contribution of this work lies in the method presented
rather than in these particular new bounds. This method allows us,
in some sense, to directly analyse prioritised algorithms in
regular graphs, so that the class of locally greedy algorithms, or
slight modifications thereof, may be applied to a wider range of
problems in regular graphs with large girth
Digraph Complexity Measures and Applications in Formal Language Theory
We investigate structural complexity measures on digraphs, in particular the
cycle rank. This concept is intimately related to a classical topic in formal
language theory, namely the star height of regular languages. We explore this
connection, and obtain several new algorithmic insights regarding both cycle
rank and star height. Among other results, we show that computing the cycle
rank is NP-complete, even for sparse digraphs of maximum outdegree 2.
Notwithstanding, we provide both a polynomial-time approximation algorithm and
an exponential-time exact algorithm for this problem. The former algorithm
yields an O((log n)^(3/2))- approximation in polynomial time, whereas the
latter yields the optimum solution, and runs in time and space O*(1.9129^n) on
digraphs of maximum outdegree at most two. Regarding the star height problem,
we identify a subclass of the regular languages for which we can precisely
determine the computational complexity of the star height problem. Namely, the
star height problem for bideterministic languages is NP-complete, and this
holds already for binary alphabets. Then we translate the algorithmic results
concerning cycle rank to the bideterministic star height problem, thus giving a
polynomial-time approximation as well as a reasonably fast exact exponential
algorithm for bideterministic star height.Comment: 19 pages, 1 figur
Ramsey-nice families of graphs
For a finite family of fixed graphs let be
the smallest integer for which every -coloring of the edges of the
complete graph yields a monochromatic copy of some . We
say that is -nice if for every graph with
and for every -coloring of there exists a
monochromatic copy of some . It is easy to see that if
contains no forest, then it is not -nice for any . It seems
plausible to conjecture that a (weak) converse holds, namely, for any finite
family of graphs that contains at least one forest, and for all
(or at least for infinitely many values of ),
is -nice. We prove several (modest) results in support of this
conjecture, showing, in particular, that it holds for each of the three
families consisting of two connected graphs with 3 edges each and observing
that it holds for any family containing a forest with at most 2
edges. We also study some related problems and disprove a conjecture by
Aharoni, Charbit and Howard regarding the size of matchings in regular
3-partite 3-uniform hypergraphs.Comment: 20 pages, 2 figure
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