12,463 research outputs found
Treewidth: Computational Experiments.
Many NP-complete graph problems can be solved in polynomial time for graphs with bounded treewidth. Equivalent results are known for pathwidth and branchwidth. In recent years, several studies have shown that this result is not only of theoretical interest but can successfully be applied to find (almost) optimal solutions or lower bounds for diverse optimization problems. To apply a tree decomposition approach, the treewidth of the graph has to be determined, independently of the application at hand. Although for fixed k, linear time algorithms exist to solve the decision problem ``treewidth at most k, their practical use is very limited. The computational tractability of treewidth has been rarely studied so far. In this paper, we compare four heuristics and two lower bounds for instances from applications such as the frequency assignment problem and the vertex coloring problem. Three of the heuristics are based on well-known algorithms to recognize triangulated graphs. The fourth heuristic recursively improves a tree decomposition by the computation of minimal separating vertex sets in subgraphs. Lower bounds can be computed from maximal cliques and the minimum degree of induced subgraphs. A computational analysis shows that the treewidth of several graphs can be identified by these methods. For other graphs, however, more sophisticated techniques are necessary.operations research and management science;
Kocay's lemma, Whitney's theorem, and some polynomial invariant reconstruction problems
Given a graph G, an incidence matrix N(G) is defined for the set of distinct
isomorphism types of induced subgraphs of G. If Ulam's conjecture is true, then
every graph invariant must be reconstructible from this matrix, even when the
graphs indexing the rows and the columns of N(G) are unspecified. It is proved
that the characteristic polynomial, the rank polynomial, and the number of
spanning trees of a graph are reconstructible from its N-matrix. These results
are stronger than the original results of Tutte in the sense that actual
subgraphs are not used. It is also proved that the characteristic polynomial of
a graph with minimum degree 1 can be computed from the characteristic
polynomials of all its induced proper subgraphs. The ideas in Kocay's lemma
play a crucial role in most proofs. Here Kocay's lemma is used to prove
Whitney's subgraph expansion theorem in a simple manner. The reconstructibility
of the characteristic polynomial is then demonstrated as a direct consequence
of Whitney's theorem as formulated here.Comment: 31 page
Proving properties of the edge elimination polynomial using equivalent graph polynomials
Averbouch, Godlin and Makowsky define the edge elimination polynomial of a
graph by a recurrence relation with respect to the deletion, contraction and
extraction of an edge. It generalizes some well-known graph polynomials such as
the chromatic polynomial and the matching polynomial. By introducing two
equivalent graph polynomials, one enumerating subgraphs and the other
enumerating colorings, we show that the edge elimination polynomial of a simple
graph is reconstructible from its polynomial deck and that it encodes the
degree sequence of an arbitrary graph.Comment: 14 pages, 1 figur
Enumerating consistent subgraphs of directed acyclic graphs: an insight into biomedical ontologies
Modern problems of concept annotation associate an object of interest (gene,
individual, text document) with a set of interrelated textual descriptors
(functions, diseases, topics), often organized in concept hierarchies or
ontologies. Most ontologies can be seen as directed acyclic graphs, where nodes
represent concepts and edges represent relational ties between these concepts.
Given an ontology graph, each object can only be annotated by a consistent
subgraph; that is, a subgraph such that if an object is annotated by a
particular concept, it must also be annotated by all other concepts that
generalize it. Ontologies therefore provide a compact representation of a large
space of possible consistent subgraphs; however, until now we have not been
aware of a practical algorithm that can enumerate such annotation spaces for a
given ontology. In this work we propose an algorithm for enumerating consistent
subgraphs of directed acyclic graphs. The algorithm recursively partitions the
graph into strictly smaller graphs until the resulting graph becomes a rooted
tree (forest), for which a linear-time solution is computed. It then combines
the tallies from graphs created in the recursion to obtain the final count. We
prove the correctness of this algorithm and then apply it to characterize four
major biomedical ontologies. We believe this work provides valuable insights
into concept annotation spaces and predictability of ontological annotation.Comment: 18 pages, 6 figure
Some graph properties determined by edge zeta functions
Stark and Terras introduced the edge zeta function of a finite graph in 1996.
The edge zeta function is the reciprocal of a polynomial in twice as many
variables as edges in the graph and can be computed in polynomial time. We look
at graph properties which we can determine using the edge zeta function. In
particular, the edge zeta function is enough to deduce the clique number, the
number of Hamiltonian cycles, and whether a graph is perfect or chordal.
Actually computing these properties takes exponential time. Finally, we present
a new example illustrating that the Ihara zeta function cannot necessarily do
the same.Comment: 15 pages, 3 figures, 26 reference
A Local Prime Factor Decomposition Algorithm for Strong Product Graphs
This work is concerned with the prime factor decomposition (PFD) of strong
product graphs. A new quasi-linear time algorithm for the PFD with respect to
the strong product for arbitrary, finite, connected, undirected graphs is
derived. Moreover, since most graphs are prime although they can have a
product-like structure, also known as approximate graph products, the practical
application of the well-known "classical" prime factorization algorithm is
strictly limited. This new PFD algorithm is based on a local approach that
covers a graph by small factorizable subgraphs and then utilizes this
information to derive the global factors. Therefore, we can take advantage of
this approach and derive in addition a method for the recognition of
approximate graph products
Supersaturation of : from Zarankiewicz towards Erd\H{o}s-Simonovits-Sidorenko
For a positive integer , a graph and a bipartite graph let denote the number of copies of in , and let
denote the minimum number of copies of in all graphs
with edges. The study of such a function is the
subject of theorems of supersaturated graphs and closely related to the
Sidorenko-Erd\H{o}s-Simonovits conjecture as well. In the present paper we
investigate the case when and in particular the quadrilateral
graph case. For , we obtain exact results if and the corresponding
Zarankiewicz number differ by at most , by a finite geometric construction
of almost difference sets. if and the corresponding
Zarankiewicz number differs by we prove asymptotically sharp
results. We also study stability questions and point out the connections to
covering and packing block designs
A Simple Explanation for the Reconstruction of Graphs
The graph reconstruction conjecture states that all graphs on at least three
vertices are determined up to isomorphism by their deck. In this paper, a
general framework for this problem is proposed to simply explain the
reconstruction of graphs. Here, we do not prove or reject the reconstruction
conjecture. But, we explain why a graph is reconstructible. For instance, the
reconstruction of small graphs which have been shown by computer, is explained
in this framework. We show that any non-regular graph has a proper induced
subgraph which is unique due to either its structure or the way of its
connection to the rest of the graph. Here, the former subgraph is defined an
anchor and the latter a connectional anchor, if it is distinguishable in the
deck. We show that if a graph has an orbit with at least three vertices whose
removal leaves an anchor, or it has two vertices whose removal leaves an anchor
with the mentioned condition in the paper, then it is reconstructible. This
simple statement can easily explain the reconstruction of a graph from its
deck
Spectra and spectral correlations of microwave graphs with symplectic symmetry
Following an idea by Joyner et al. [EPL, 107 (2014) 50004] a microwave graph
with antiunitary symmetry T obeying T^2=-1 has been realized. The Kramers
doublets expected for such systems have been clearly identified and could be
lifted by a perturbation which breaks the antiunitary symmetry. The observed
spectral level spacings distribution of the Kramers doublets is in agreement
with the predictions from the Gaussian symplectic ensemble (GSE), expected for
chaotic systems with such a symmetry. In addition results on the two-point
correlation function, the spectral form factor, the number variance and the
spectral rigidity are presented, as well as on the transition from GSE to GOE
statistics by continuously changing T from T^2=-1 to T^2=1.Comment: 14 pages, 12 figure
On applications of Razborov's flag algebra calculus to extremal 3-graph theory
In this paper, we prove several new Tur\'an density results for 3-graphs with
independent neighbourhoods. We show:
\pi(K_4^-, C_5, F_{3,2})=12/49, \pi(K_4^-, F_{3,2})=5/18, and
\pi(J_4, F_{3,2})=\pi(J_5, F_{3,2})=3/8, where J_t is the 3-graph consisting
of a single vertex x together with a disjoint set A of size t and all
3-edges containing x.
We also prove two Tur\'an density results where we forbid certain induced
subgraphs:
\pi(F_{3,2}, induced K_4^-)=3/8 and
\pi(K_5, 5-set spanning 8 edges)=3/4.
The latter result is an analogue for K_5 of Razborov's result that
\pi(K_4, 4-set spanning 1 edge)=5/9.
We give several new constructions, conjectures and bounds for Tur\'an
densities of 3-graphs which should be of interest to researchers in the area.
Our main tool is `Flagmatic', an implementation of Razborov's flag algebra
calculus, which we are making publicly available. In a bid to make the power of
Razborov's method more widely accessible, we have tried to make Flagmatic as
user-friendly as possible, hoping to remove thereby the major hurdle that needs
to be cleared before using the flag algebra calculus.
Finally, we spend some time reflecting on the limitations of our approach,
and in particular on which problems we may be unable to solve. Our discussion
of the `complexity barrier' for the flag algebra calculus may be of general
interest.Comment: 31 pages, 5 figures; version 2 corrects some minor mistake
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