Showing papers on "Path graph published in 2002"
TL;DR: In this article, it was shown that the largest possible function f(e, H) is polynomial in e if and only if H is bipartite, which implies that there is a one-sided error property tester for checking H-freeness, whose query complexity is O(1/e) if H are bipartites.
Abstract: Let H be a fixed graph with h vertices, let G be a graph on n vertices, and suppose that at least en2 edges have to be deleted from it to make it H-free. It is known that in this case G contains at least f(e, H)nh copies of H. We show that the largest possible function f(e, H) is polynomial in e if and only if H is bipartite. This implies that there is a one-sided error property tester for checking H-freeness, whose query complexity is polynomial in 1/e, if and only if H is bipartite.
181 citations
Posted Content•
TL;DR: A theory of graph C -algebras using path groupoids and inverse semigroups was developed in this article.However, it is not assumed that the theory applies to graphs for which there are vertices emitting a countably infinite set of edges.
Abstract: We develop a theory of graph C -algebras using path groupoids and inverse semigroups. Row finiteness is not assumed so that the theory applies to graphs for which there are vertices emitting a countably infinite set of edges. We show that the path groupoid is amenable, and give a groupoid proof of a recent theorem of Szymanski characterizing when a graph C - algebra is simple.
129 citations
TL;DR: In this paper, it was shown that the minimum number of distinct eigenvalues of an Hermitian matrix whose graph is T (diagonal entries free) is at least the number of vertices in a longest path of T. This is another step toward the general problem of characterizing the possible multiplicities for a given graph.
Abstract: It is shown that for any tree T the minimum number of distinct eigenvalues of an Hermitian matrix whose graph is T (diagonal entries free) is at least the number of vertices in a longest path of T. This is another step toward the general problem of characterizing the possible multiplicities for a given graph. Related observations are made and the result facilitates a table of multiplicities for trees on fewer than 8 vertices.
68 citations
TL;DR: Improved semidefinite programming based approximation algorithms are obtained for all the natural maximum bisection problems of graphs and improve, extend and unify results of Frieze and Jerrum, Feige and Langberg, Ye, and others.
Abstract: We obtain improved semidefinite programming based approximation algorithms for all the natural maximum bisection problems of graphs. Among the problems considered are: MAX n/2-BISECTION--partition the vertices of the graph into two sets of equal size such that the total weight of edges connecting vertices from different sides is maximized; MAX n/2-VERTEX-COVER--find a set containing half of the vertices such that the total weight of edges touching this set is maximized; MAX n/2-DENSE-SUBGRAPH--find a set containing half of the vertices such that the total weight of edges connecting two vertices from this set is maximized; and MAX n/2-UNCUT partition the vertices into two sets of equal size such that the total weight of edges that do not cross the cut is maximized. We also consider the directed versions of these problems, such as MAX n/2-DIRECTED-BISECTION and MAX n/2-DIRECTED-UNCUT. These results can be used to obtain improved approximation algorithms for the unbalanced versions of the partition problems mentioned above, where we want to partition the graph into two sets of size k and n - k, where k is not necessarily n/2. Our results improve, extend and unify results of Frieze and Jerrum, Feige and Langberg, Ye, and others. All these results may be viewed as extensions of the MAX CUT algorithm of Goemans and Williamson, and the MAX 2-SAT and MAX DI-CUT algorithms of Feige and Goemans.
66 citations
TL;DR: In this article, it was proved that a graph whose (0, 1) -adjacency matrix has the spectrum of P n, the complement of the path on n vertices, must be P n.
63 citations
Journal Article•
TL;DR: The structure of plane graphs in terms of stars and bunches was studied in this article, where it was shown that a plane graph contains a $(d-1)$-star centred at a vertex of degree $d\leq5$ and the sum of the degrees of the vertices in the star is bounded.
Abstract: Given a plane graph, a $k$-star at $u$ is a set of $k$ vertices with a common neighbour $u$; and a bunch is a maximal collection of paths of length at most two in the graph, such that all paths have the same end vertices and the edges of the paths form consecutive edges (\,in the natural order in the plane graph\,) around the two end vertices. We first prove a theorem on the structure of plane graphs in terms of stars and bunches. The result states that a plane graph contains a $(d-1)$-star centred at a vertex of degree $d\leq5$ and the sum of the degrees of the vertices in the star is bounded, or there exists a large bunch.
62 citations
06 Jan 2002
TL;DR: This work examines the problem of finding and representing possible layouts of the intervals, given a probe interval graph, and obtains an O(n + m) bound, where n is the number of vertices and m is thenumber of edges.
Abstract: An interval graph for a set of intervals on a line consists of one vertex for each interval, and an edge for each pair of intersecting intervals. A probe interval graph is obtained from an interval graph by designating a subset P of vertices as probes, and removing the edges between pairs of vertices in the remaining set N of non-probes. We examine the problem of finding and representing possible layouts of the intervals, given a probe interval graph. We obtain an O(n + m log n) bound, where n is the number of vertices and m is the number of edges. The problem is motivated by an application to molecular biology.
60 citations
TL;DR: The induced path interval J(u,v) consists of the vertices on the induced paths between u and v in a connected graph G, in which the induced path intervals define a proper betweenness.
49 citations
TL;DR: In this paper, the authors determined the structure of the chemical trees (the graph representation of acyclic saturated hydrocarbons) that are extremal with respect to the Wiener index W, the largest graph eigenvalue λ 1, the connectivity index X, the graph energy E and the Hosoya index Z.
Abstract: A variety of molecular-graph-based structure-descriptors were proposed, in particular the Wiener index W, the largest graph eigenvalue λ 1 , the connectivity index X, the graph energy E and the Hosoya index Z, capable of measuring the branching of the carbon-atom skeleton of organic compounds, and therefore suitable for describing several of their physico-chemical properties We now determine the structure of the chemical trees (= the graph representation of acyclic saturated hydrocarbons) that are extremal with respect to W, λ 1 , E, and Z, whereas the analogous problem for x was solved earlier Among chemical trees with 5, 6, 7, and 3k + 2 vertices, k = 2, 3,, one and the same tree has maximum λ 1 and minimum W, E, Z Among chemical trees with 3k and 3k + 1 vertices, k = 3, 4,, one tree has minimum W and maximum λ 1 and another minimum E and Z
40 citations
TL;DR: It is shown that the minimum number of edges of a graph that contains every graph with n edges as a subgraph is Δ(n2/(log2n); this improves a result of Babai, Chung, Erdos, Graham and Spencer.
39 citations
TL;DR: This work develops a pseudo-polynomial algorithm for (both positive and negative) integer weights, and shows how to find the minimum absolute cost of any path between two given nodes in a graph with integer weights in O(W2n3) time.
TL;DR: It is shown that for every triple l,k,n of integers with n≥3 and 2≤l≤k≤n−1, there exists a noncomplete connected graph G of order n with ω(G))=l and con(G)=k.
Abstract: For two vertices u and v of a connected graph G, the set I[u,v] consists of all those vertices lying on a u−v shortest path in G, while for a set S of vertices of G, the set I[S] is the union of all sets I[u,v] for u,v∈S. A set S is convex if I[S]=S. The convexity number con(G) of G is the maximum cardinality of a proper convex set of G. The clique number ω(G) is the maximum cardinality of a clique in G. If G is a connected graph of order n that is not complete, then n≥3 and 2≤ω(G)≤con(G)≤n−1. It is shown that for every triple l,k,n of integers with n≥3 and 2≤l≤k≤n−1, there exists a noncomplete connected graph G of order n with ω(G)=l and con(G)=k. Other results on convex numbers are also presented.
TL;DR: This paper presents efficient algorithms for finding the l-core of a tree using two different types of recursive principle, and presents an O(nl) time algorithm and a procedure with time complexity of O(nlog2n).
03 Jul 2002
TL;DR: This work gives algorithms that solve the clustering problem with high probability and can handle O(?n/ log n) clusters in an n-vertex graph, while all previous algorithms require that the number of clusters is constant.
Abstract: The following probabilistic process models the generation of noisy clustering data: Clusters correspond to disjoint sets of vertices in a graph. Each two vertices from the same set are connected by an edge with probability p, and each two vertices from different sets are connected by an edge with probability r < p. The goal of the clustering problem is to reconstruct the clusters from the graph. We give algorithms that solve this problem with high probability. Compared to previous studies, our algorithms have lower time complexity and wider parameter range of applicability. In particular, our algorithms can handle O(?n/ log n) clusters in an n-vertex graph, while all previous algorithms require that the number of clusters is constant.
TL;DR: The graphs in which, with respect to any nonnegative vertex weights, median sets always induce connected subgraphs are characterized.
Abstract: The median set of a graph G with weighted vertices comprises the vertices minimizing the average weighted distance to the vertices of G. We characterize the graphs in which, with respect to any nonnegative vertex weights, median sets always induce connected subgraphs. The characteristic conditions can be tested in polynomial time (by employing linear programming) and are immediately verified for a number of specific graph classes.
TL;DR: A new graph metric, in contrast to the standard geodesic distance in connected graphs, it takes into account all paths between vertices, and the notion of a cumulative weight of connections between two vertices is proposed.
TL;DR: It is proved that if G is a claw-free graph with minimum degree at least d, then G has a path factor such that the order of each path is at leastd+1.
TL;DR: In this paper, the authors give a linear-time algorithm for finding a minimum path cover for a tree that uses a reduction to a maximum flow problem, and show how to extend the reduction to construct, if possible, a covering of the vertices of a bipartite graph with vertex-disjoint cycles.
Abstract: A minimum Hamiltonian completion of a graph G is a minimum-size set of edges that, when added to G, guarantee a Hamiltonian path. Finding a Hamiltonian completion has applications to frequency assignment as well as distributed computing. If the new edges are deleted from the Hamiltonian path, one is left with a minimum path cover, a minimum-size set of vertex-disjoint paths that cover the vertices of G. For arbitrary graphs, constructing a minimum Hamiltonian completion or path cover is clearly NP-hard, but there exists a linear-time algorithm for trees. In this paper we first give a description and proof of correctness for this linear-time algorithm that is simpler and more intuitive than those given previously. We show that the algorithm extends also to unicyclic graphs. We then give a new method for finding an optimal path cover or Hamiltonian completion for a tree that uses a reduction to a maximum flow problem. In addition, we show how to extend the reduction to construct, if possible, a covering of the vertices of a bipartite graph with vertex-disjoint cycles, that is, a 2-factor.
TL;DR: By introducing a general class of dynamical graphs, the model provides a bridge between two well-known models and is a unified overview on rather different models and the relations between them.
Abstract: We study the large-time dynamics of a Markov process whose states are finite but unbounded graphs. The number of vertices is described by a supercritical branching process, and the edges follow a certain mean-field dynamics determined by the rates of appending and deleting: the older an edge is, the lesser is the probability that it is still in the graph. The lifetime of any edge is distributed exponentially. We call its mean value (common for all edges) a parameter of memory, since it shows for how long the system keeps a particular connection between the vertices in the graph. We show that our model provides a bridge between two well-known models: when the parameter of memory goes to infinity this is a generalized model of random growth, and when this parameter is zero, i.e., no memory, our model behaves as a random graph. Thus by introducing a general class of dynamical graphs we have a unified overview on rather different models and the relations between them. We find all the critical values of the parameters at which our model exhibits phase transitions and describe the properties of the phase diagram. Finally, we compare and discuss the efficiency of the corresponding networks.
TL;DR: By using the so-defined circuit/path transformations together with an edge-switching method, the following conjectures are proved: the edges of a connected graph on n vertices can be covered by at most ?
TL;DR: It is proved that there exists a function a:N0×R+?N such that if G is a 4-connected graph of order n embedded on a surface of Euler genus g such that the face-width of G is at least a(g, ?), then G can be covered by two cycles each of which has length at least (1??)n.
Journal Article•
TL;DR: The structure of plane triangulations in terms of stars and bunches has been studied in this article, where it is shown that a plane triangle has at most one star centred at a vertex of degree n ≥ 5 and the sum of the vertices in the star is bounded.
Abstract: Given a plane graph, a $k$-star at $u$ is a set of $k$ vertices with a common neighbour $u$; and a bunch is a maximal collection of paths of length at most two in the graph, such that all paths have the same end vertices and the edges of the paths form consecutive edges (\,in the natural order in the plane graph\,) around the two end vertices. We prove a theorem on the structure of plane triangulations in terms of stars and bunches. The result states that a plane triangulation contains a $(d-1)$-star centred at a vertex of degree $d\leq5$ and the sum of the degrees of the vertices in the star is bounded, or there exists a large bunch.
Posted Content•
TL;DR: In this article, it was shown that for c ≥ 2.4682, a random graph on n vertices with c n (1+o(1)) edges almost surely has no 3-colouring.
Abstract: We show that for c >= 2.4682, a random graph on n vertices with c n (1+o(1)) edges almost surely has no 3-colouring. This improves on the current best upper bound of 2.4947.
TL;DR: In this paper, it was shown that the number of connected components of a finite -solvable group is at most 2, and when disconnected, each of the two components is a complete graph.
Abstract: Let be a finite -solvable group. Attach to the following graph : its vertices are the non-central conjugacy classes of -regular elements of , and two vertices are connected by an edge if their cardinalities are not coprime. We prove that the number of connected components of is at most 2. When is connected, then the diameter of the graph is at most 3, and when is disconnected, then each of the two components is a complete graph.
TL;DR: It is proved that in every case this set contains all the integers between 3 and some t, where t depends on the order of the graph and the degrees of vertices.
TL;DR: In this paper, it was shown that a bipartite distance-regular graph is a tail if and only if one of the following holds: (i) Δ is a path; (ii) Δ has two connected components, each of which is a clique; (iii) D = 6; and (iv) D ≥ 3.
TL;DR: A simple embedding technique is used to find exact formulae for the edge–isoperimetric number and the bisection width of a cylinder in certain cases, e.g. when the size of the largest factor is even.
Patent•
22 Mar 2002TL;DR: In this article, the simplification of a surface represented as a height field is discussed, resulting in a triangular mesh having n vertices, such that the topology of the surface is maintained in the simplified surface representation.
Abstract: Simplifying a surface represented as a height field. The simplification resulting in a triangular mesh having n vertices, such that the topology of the surface is maintained in the simplified surface representation. The simplification including assigning error values to each vertex not having a missing value, identifying as required vertices those vertices that may not be removed without altering the topology of the surface; and (1) if n is less than or equal the number of required vertices, remove all other vertices or (2) if n is greater than or equal the number of required vertices, remove u-n vertices, where u is the number of vertices that do not have missing values in the original height field.
TL;DR: It is shown that in a 4-connected maximal planar graph there is a cycle in the graph that contains the four vertices and visits them in the order a, b, c and d.
TL;DR: This work presents the first (linear time) 2-approximation algorithm for the edge clique complement problem, and uses the local-ratio technique to develop 2- approximation algorithms for the first two variants of the problem.