461 research outputs found
Universal graphs with forbidden subgraphs and algebraic closure
We apply model theoretic methods to the problem of existence of countable
universal graphs with finitely many forbidden connected subgraphs. We show that
to a large extent the question reduces to one of local finiteness of an
associated''algebraic closure'' operator. The main applications are new
examples of universal graphs with forbidden subgraphs and simplified treatments
of some previously known cases
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
A look at cycles containing specified elements of a graph
AbstractThis article is intended as a brief survey of problems and results dealing with cycles containing specified elements of a graph. It is hoped that this will help researchers in the area to identify problems and areas of concentration
Hamiltonicity, Pancyclicity, and Cycle Extendability in Graphs
The study of cycles, particularly Hamiltonian cycles, is very important in many applications.
Bondy posited his famous metaconjecture, that every condition sufficient for Hamiltonicity actually guarantees a graph is pancyclic. Pancyclicity is a stronger structural property than Hamiltonicity.
An even stronger structural property is for a graph to be cycle extendable. Hendry conjectured that any graph which is Hamiltonian and chordal is cycle extendable.
In this dissertation, cycle extendability is investigated and generalized. It is proved that chordal 2-connected K1,3-free graphs are cycle extendable. S-cycle extendability was defined by Beasley and Brown, where S is any set of positive integers. A conjecture is presented that Hamiltonian chordal graphs are {1, 2}-cycle extendable.
Dirac’s Theorem is an classic result establishing a minimum degree condition for a graph to be Hamiltonian. Ore’s condition is another early result giving a sufficient condition for Hamiltonicity. In this dissertation, generalizations of Dirac’s and Ore’s Theorems are presented.
The Chvatal-Erdos condition is a result showing that if the maximum size of an independent set in a graph G is less than or equal to the minimum number of vertices whose deletion increases the number of components of G, then G is Hamiltonian. It is proved here that the Chvatal-Erdos condition guarantees that a graph is cycle extendable. It is also shown that a graph having a Hamiltonian elimination ordering is cycle extendable.
The existence of Hamiltonian cycles which avoid sets of edges of a certain size and certain subgraphs is a new topic recently investigated by Harlan, et al., which clearly has applications to scheduling and communication networks among other things. The theory is extended here to bipartite graphs. Specifically, the conditions for the existence of a Hamiltonian cycle that avoids edges, or some subgraph of a certain size, are determined for the bipartite case.
Briefly, this dissertation contributes to the state of the art of Hamiltonian cycles, cycle extendability and edge and graph avoiding Hamiltonian cycles, which is an important area of graph theory
Three Existence Problems in Extremal Graph Theory
Proving the existence or nonexistence of structures with specified properties is the impetus for many classical results in discrete mathematics.
In this thesis we take this approach to three different structural questions rooted in extremal graph theory.
When studying graph representations, we seek efficient ways to encode the structure of a graph.
For example, an {\it interval representation} of a graph is an assignment of intervals on the real line to the vertices of such that two vertices are adjacent if and only if their intervals intersect.
We consider graphs that have {\it bar -visibility representations}, a generalization of both interval representations and another well-studied class of representations known as visibility representations.
We obtain results on , the family of graphs having bar -visibility representations.
We also study .
In particular, we determine the largest complete graph having a bar -visibility representation, and we show that there are graphs that do not have bar -visibility representations for any .
Graphs arise naturally as models of networks, and there has been much study of the movement of information or resources in graphs.
Lampert and Slater \cite{LS} introduced {\it acquisition} in weighted graphs, whereby weight moves around provided that each move transfers weight from a vertex to a heavier neighbor.
Our goal in making acquisition moves is to consolidate all of the weight in on the minimum number of vertices; this minimum number is the {\it acquisition number} of .
We study three variations of acquisition in graphs: when a move must transfer all the weight from a vertex to its neighbor, when each move transfers a single unit of weight, and when a move can transfer any positive amount of weight.
We consider acquisition numbers in various families of graphs, including paths, cycles, trees, and graphs with diameter .
We also study, under the various acquisition models, those graphs in which all the weight can be moved to a single vertex.
Restrictive local conditions often have far-reaching impacts on the global structure of mathematical objects.
Some local conditions are so limiting that very few objects satisfy the requirements.
For example, suppose that we seek a graph in which every two vertices have exactly one common neighbor.
Such graphs are called {\it friendship graphs}, and Wilf~\cite{Wilf} proved that the only such graphs consist of edge-disjoint triangles sharing a common vertex.
We study a related structural restriction where similar phenomena occur.
For a fixed graph , we consider those graphs that do not contain and such that the addition of any edge completes exactly one copy of .
Such a graph is called {\it uniquely -saturated}.
We study the existence of uniquely -saturated graphs when is a path or a cycle.
In particular, we determine all of the uniquely -saturated graphs; there are exactly ten.
Interestingly, the uniquely -saturated graphs are precisely the friendship graphs characterized by Wilf
Optimal Vertex Fault Tolerant Spanners (for fixed stretch)
A -spanner of a graph is a sparse subgraph whose shortest path
distances match those of up to a multiplicative error . In this paper we
study spanners that are resistant to faults. A subgraph is an
vertex fault tolerant (VFT) -spanner if is a -spanner
of for any small set of vertices that might "fail." One
of the main questions in the area is: what is the minimum size of an fault
tolerant -spanner that holds for all node graphs (as a function of ,
and )? This question was first studied in the context of geometric
graphs [Levcopoulos et al. STOC '98, Czumaj and Zhao SoCG '03] and has more
recently been considered in general undirected graphs [Chechik et al. STOC '09,
Dinitz and Krauthgamer PODC '11].
In this paper, we settle the question of the optimal size of a VFT spanner,
in the setting where the stretch factor is fixed. Specifically, we prove
that every (undirected, possibly weighted) -node graph has a
-spanner resilient to vertex faults with edges, and this is fully optimal (unless the famous Erdos Girth
Conjecture is false). Our lower bound even generalizes to imply that no data
structure capable of approximating similarly can
beat the space usage of our spanner in the worst case. We also consider the
edge fault tolerant (EFT) model, defined analogously with edge failures rather
than vertex failures. We show that the same spanner upper bound applies in this
setting. Our data structure lower bound extends to the case (and hence we
close the EFT problem for -approximations), but it falls to for . We leave it as an open problem to
close this gap.Comment: To appear in SODA 201
Complexity Framework for Forbidden Subgraphs II: When Hardness Is Not Preserved under Edge Subdivision
For a fixed set of graphs, a graph is -subgraph-free
if does not contain any as a (not necessarily induced)
subgraph. A recently proposed framework gives a complete classification on
-subgraph-free graphs (for finite sets ) for problems that
are solvable in polynomial time on graph classes of bounded treewidth,
NP-complete on subcubic graphs, and whose NP-hardness is preserved under edge
subdivision. While a lot of problems satisfy these conditions, there are also
many problems that do not satisfy all three conditions and for which the
complexity -subgraph-free graphs is unknown.
In this paper, we study problems for which only the first two conditions of
the framework hold (they are solvable in polynomial time on classes of bounded
treewidth and NP-complete on subcubic graphs, but NP-hardness is not preserved
under edge subdivision). In particular, we make inroads into the classification
of the complexity of four such problems: -Induced Disjoint Paths,
-Colouring, Hamilton Cycle and Star -Colouring. Although we do not
complete the classifications, we show that the boundary between polynomial time
and NP-complete differs among our problems and differs from problems that do
satisfy all three conditions of the framework. Hence, we exhibit a rich
complexity landscape among problems for -subgraph-free graph classes
- …