10 research outputs found
Tree Graphs and Orthogonal Spanning Tree Decompositions
Given a graph G, we construct T(G), called the tree graph of G. The vertices of T(G) are the spanning trees of G, with edges between vertices when their respective spanning trees differ only by a single edge. In this paper we detail many new results concerning tree graphs, involving topics such as clique decomposition, planarity, and automorphism groups. We also investigate and present a number of new results on orthogonal tree decompositions of complete graphs
Finding Cycles and Trees in Sublinear Time
We present sublinear-time (randomized) algorithms for finding simple cycles
of length at least and tree-minors in bounded-degree graphs. The
complexity of these algorithms is related to the distance of the graph from
being -minor-free (resp., free from having the corresponding tree-minor).
In particular, if the graph is far (i.e., -far) {from} being
cycle-free, i.e. if one has to delete a constant fraction of edges to make it
cycle-free, then the algorithm finds a cycle of polylogarithmic length in time
\tildeO(\sqrt{N}), where denotes the number of vertices. This time
complexity is optimal up to polylogarithmic factors.
The foregoing results are the outcome of our study of the complexity of {\em
one-sided error} property testing algorithms in the bounded-degree graphs
model. For example, we show that cycle-freeness of -vertex graphs can be
tested with one-sided error within time complexity
\tildeO(\poly(1/\e)\cdot\sqrt{N}). This matches the known
query lower bound, and contrasts with the fact that any minor-free property
admits a {\em two-sided error} tester of query complexity that only depends on
the proximity parameter \e. For any constant , we extend this result
to testing whether the input graph has a simple cycle of length at least .
On the other hand, for any fixed tree , we show that -minor-freeness has
a one-sided error tester of query complexity that only depends on the proximity
parameter \e.
Our algorithm for finding cycles in bounded-degree graphs extends to general
graphs, where distances are measured with respect to the actual number of
edges. Such an extension is not possible with respect to finding tree-minors in
complexity.Comment: Keywords: Sublinear-Time Algorithms, Property Testing, Bounded-Degree
Graphs, One-Sided vs Two-Sided Error Probability Updated versio
Local properties of graphs
We say a graph is locally P if the induced graph on the neighbourhood of every vertex has the property P. Specically, a graph is locally traceable (LT) or locally hamiltonian (LH) if the induced graph on the neighbourhood of every vertex is traceable or hamiltonian, respectively. A locally locally hamiltonian (L2H) graph is a graph in which the graph induced by the neighbourhood of each vertex is an
LH graph. This concept is generalized to an arbitrary degree of nesting, to make it possible to work with LkH graphs. This thesis focuses on the global cycle properties of LT, LH and LkH graphs. Methods are developed to construct and combine such graphs to create others with desired properties. It is shown that with the exception of three graphs, LT graphs with maximum degree no greater than 5 are fully cycle extendable (and hence hamiltonian), but
the Hamilton cycle problem for LT graphs with maximum degree 6 is NP-complete. Furthermore, the smallest nontraceable LT graph has order 10, and the smallest value of the maximum degree for which LT graphs can be nontraceable is 6. It is also shown that LH graphs with maximum degree 6 are fully cycle extendable, and that there exist nonhamiltonian LH graphs with maximum degree 9 or less for all orders greater than 10. The Hamilton cycle problem is shown to be
NP-complete for LH graphs with maximum degree 9. The construction of r-regular nonhamiltonian graphs is demonstrated, and it is shown that the number of vertices in a longest path in an LH graph can contain a vanishing fraction of the vertices of the graph. NP-completeness of the Hamilton cycle problem for LkH graphs for higher values of k is also investigated.Mathematical SciencesD. Phil. (Mathematics
Hamiltonian and Pancyclic Graphs in the Class of Self-Centered Graphs with Radius Two
The paper deals with Hamiltonian and pancyclic graphs in the class of all self-centered graphs of radius 2. For both of the two considered classes of graphs we have done the following. For a given number n of vertices, we have found an upper bound of the minimum size of such graphs. For n ≤ 12 we have found the exact values of the minimum size. On the other hand, the exact value of the maximum size has been found for every n. Moreover, we have shown that such a graph (of order n and) of size m exists for every m between the minimum and the maximum size. For n ≤ 10 we have found all nonisomorphic graphs of the minimum size, and for n = 11 only for Hamiltonian graphs
Hamiltonian and Pancyclic Graphs in the Class of Self-Centered Graphs with Radius Two
The paper deals with Hamiltonian and pancyclic graphs in the class of all self-centered graphs of radius 2. For both of the two considered classes of graphs we have done the following. For a given number n of vertices, we have found an upper bound of the minimum size of such graphs. For n ≤ 12 we have found the exact values of the minimum size. On the other hand, the exact value of the maximum size has been found for every n. Moreover, we have shown that such a graph (of order n and) of size m exists for every m between the minimum and the maximum size. For n ≤ 10 we have found all nonisomorphic graphs of the minimum size, and for n = 11 only for Hamiltonian graphs
SIMULATING SEISMIC WAVE PROPAGATION IN TWO-DIMENSIONAL MEDIA USING DISCONTINUOUS SPECTRAL ELEMENT METHODS
We introduce a discontinuous spectral element method for simulating seismic wave in 2- dimensional elastic media. The methods combine the flexibility of a discontinuous finite
element method with the accuracy of a spectral method. The elastodynamic equations are discretized using high-degree of Lagrange interpolants and integration over an element is
accomplished based upon the Gauss-Lobatto-Legendre integration rule. This combination of discretization and integration results in a diagonal mass matrix and the use of discontinuous finite element method makes the calculation can be done locally in each element. Thus, the algorithm is simplified drastically. We validated the results of one-dimensional problem by comparing them with finite-difference time-domain method and exact solution. The comparisons show excellent agreement