22,153 research outputs found
On Perfect Matchings in Matching Covered Graphs
Let be a matching-covered graph, i.e., every edge is contained in a
perfect matching. An edge subset of is feasible if there exists two
perfect matchings and such that . Lukot'ka and Rollov\'a proved that an edge subset of a regular
bipartite graph is not feasible if and only if is switching-equivalent to
, and they further ask whether a non-feasible set of a regular graph
of class 1 is always switching-equivalent to either or ? Two
edges of are equivalent to each other if a perfect matching of
either contains both of them or contains none of them. An equivalent class of
is an edge subset with at least two edges such that the edges of
are mutually equivalent. An equivalent class is not a feasible set. Lov\'asz
proved that an equivalent class of a brick has size 2. In this paper, we show
that, for every integer , there exist infinitely many -regular
graphs of class 1 with an arbitrarily large equivalent class such that
is not switching-equivalent to either or , which provides a
negative answer to the problem proposed by Lukot'ka and Rollov\'a. Further, we
characterize bipartite graphs with equivalent class, and characterize
matching-covered bipartite graphs of which every edge is removable.Comment: 10 pages, 3 figure
State transfer in strongly regular graphs with an edge perturbation
Quantum walks, an important tool in quantum computing, have been very
successfully investigated using techniques in algebraic graph theory. We are
motivated by the study of state transfer in continuous-time quantum walks,
which is understood to be a rare and interesting phenomenon. We consider a
perturbation on an edge of a graph where we add a weight to the
edge and a loop of weight to each of and . We characterize when
for this perturbation results in strongly cospectral vertices and .
Applying this to strongly regular graphs, we give infinite families of strongly
regular graphs where some perturbation results in perfect state transfer.
Further, we show that, for every strongly regular graph, there is some
perturbation which results in pretty good state transfer. We also show for any
strongly regular graph and edge , that
does not depend on the choice of .Comment: 25 page
Associated primes of powers of edge ideals and ear decompositions of graphs
In this paper, we give a complete description of the associated primes of
every power of the edge ideal in terms of generalized ear decompositions of the
graph. This result establishes a surprising relationship between two seemingly
unrelated notions of Commutative Algebra and Combinatorics. It covers all
previous major results in this topic and has several interesting consequences.Comment: 28 pages, 7 figures. This revision contains more comments on the
methods and the results of the paper. Important terminology and notations now
appear in Definition environments for the convenience of the readers. That
changes the numeration of the results considerably. This paper will appear in
Trans. Amer. Math. So
Line-graphs of cubic graphs are normal
A graph is called normal if its vertex set can be covered by cliques and also
by stable sets, such that every such clique and stable set have non-empty
intersection. This notion is due to Korner, who introduced the class of normal
graphs as an extension of the class of perfect graphs. Normality has also
relevance in information theory. Here we prove, that the line graphs of cubic
graphs are normal.Comment: 16 pages, 10 figure
Parameterized algorithm for weighted independent set problem in bull-free graphs
The maximum stable set problem is NP-hard, even when restricted to
triangle-free graphs. In particular, one cannot expect a polynomial time
algorithm deciding if a bull-free graph has a stable set of size , when
is part of the instance. Our main result in this paper is to show the existence
of an FPT algorithm when we parameterize the problem by the solution size .
A polynomial kernel is unlikely to exist for this problem. We show however that
our problem has a polynomial size Turing-kernel. More precisely, the hard cases
are instances of size . As a byproduct, if we forbid odd holes in
addition to the bull, we show the existence of a polynomial time algorithm for
the stable set problem. We also prove that the chromatic number of a bull-free
graph is bounded by a function of its clique number and the maximum chromatic
number of its triangle-free induced subgraphs. All our results rely on a
decomposition theorem of bull-free graphs due to Chudnovsky which is modified
here, allowing us to provide extreme decompositions, adapted to our
computational purpose
Pretty good state transfer on double stars
Let A be the adjacency matrix of a graph and suppose U(t)=exp(itA). We
view A as acting on \cx^{V(X)} and take the standard basis of this space to
be the vectors for in . Physicists say that we have perfect
state transfer from vertex to at time if there is a scalar
such that . (Since is unitary,
\norm\gamma=1.) For example, if is the -cube and and are at
distance then we have perfect state transfer from to at time
. Despite the existence of this nice family, it has become clear that
perfect state transfer is rare. Hence we consider a relaxation: we say that we
have pretty good state transfer from to if there is a complex number
and, for each positive real there is a time such that
\norm{U(t)e_u - \gamma e_v} < \epsilon. Again we necessarily have
.
Godsil, Kirkland, Severini and Smith showed that we have have pretty good
state transfer between the end vertices of the path if and only is
a power of two, a prime, or twice a prime. (There is perfect state transfer
between the end vertices only for and .) It is something of a
surprise that the occurrence of pretty good state transfer is characterized by
a number-theoretic condition. In this paper we study double-star graphs, which
are trees with two vertices of degree and all other vertices with degree
one. We prove that there is never perfect state transfer between the two
vertices of degree , and that there is pretty good state transfer between
them if and only if is a perfect square.Comment: 15 pages, 2 EPS figure
Polynomial-time algorithms for minimum weighted colorings of ()-free graphs and related graph classes
We design an algorithm to find a minimum weighted coloring of a
()-free graph. Furthermore, the same technique can be used to
solve the same problem for several classes of graphs, defined by forbidden
induced subgraphs, such as (diamond, co-diamond)-free graphs
A Canonical Characterization of the Family of Barriers in General Graphs
Given a graph, a barrier is a set of vertices determined by the Berge
formula---the min-max theorem characterizing the size of maximum matchings. The
notion of barriers plays important roles in numerous contexts of matching
theory, since barriers essentially coincides with dual optimal solutions of the
maximum matching problem. In a special class of graphs called the elementary
graphs, the family of maximal barriers forms a partition of the vertices; this
partition was found by Lov\'asz and is called the canonical partition. The
canonical partition has produced many fundamental results in matching theory,
such as the two ear theorem. However, in non-elementary graphs, the family of
maximal barriers never forms a partition, and there has not been the canonical
partition for general graphs. In this paper, using our previous work, we give a
canonical description of structures of the odd-maximal barriers---a class of
barriers including the maximal barriers---for general graphs; we also reveal
structures of odd components associated with odd-maximal barriers. This result
of us can be regarded as a generalization of Lov\'asz's canonical partition.Comment: 22 pages, no figure
On Brambles, Grid-Like Minors, and Parameterized Intractability of Monadic Second-Order Logic
Brambles were introduced as the dual notion to treewidth, one of the most
central concepts of the graph minor theory of Robertson and Seymour. Recently,
Grohe and Marx showed that there are graphs G, in which every bramble of order
larger than the square root of the treewidth is of exponential size in |G|. On
the positive side, they show the existence of polynomial-sized brambles of the
order of the square root of the treewidth, up to log factors. We provide the
first polynomial time algorithm to construct a bramble in general graphs and
achieve this bound, up to log-factors. We use this algorithm to construct
grid-like minors, a replacement structure for grid-minors recently introduced
by Reed and Wood, in polynomial time. Using the grid-like minors, we introduce
the notion of a perfect bramble and an algorithm to find one in polynomial
time. Perfect brambles are brambles with a particularly simple structure and
they also provide us with a subgraph that has bounded degree and still large
treewidth; we use them to obtain a meta-theorem on deciding certain
parameterized subgraph-closed problems on general graphs in time singly
exponential in the parameter.
The second part of our work deals with providing a lower bound to Courcelle's
famous theorem, stating that every graph property that can be expressed by a
sentence in monadic second-order logic (MSO), can be decided by a linear time
algorithm on classes of graphs of bounded treewidth. Using our results from the
first part of our work we establish a strong lower bound for tractability of
MSO on classes of colored graphs
On the strong chromatic index and induced matching of tree-cographs, permutation graphs and chordal bipartite graphs
We show that there exist linear-time algorithms that compute the strong
chromatic index and a maximum induced matching of tree-cographs when the
decomposition tree is a part of the input. We also show that there exist
efficient algorithms for the strong chromatic index of (bipartite) permutation
graphs and of chordal bipartite graphs.Comment: arXiv admin note: substantial text overlap with arXiv:1110.169
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