# VANDERBILT UNIV NASHVILLE TN Dept. of MATHEMATICS

Overview
Works: 21 works in 22 publications in 1 language and 22 library holdings
Publication Timeline
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Most widely held works by VANDERBILT UNIV NASHVILLE TN Dept. of MATHEMATICS
Matching Extension in Bipartite Graphs. 1. Introduction and Terminology( Book )

2 editions published in 1986 in English and held by 2 WorldCat member libraries worldwide

In the present paper, we wish to treat some relationships between toughness of a graph and the n-extendability of the graph. We will prove two results. The first says essentially that if a graph has sufficiently high toughness (and has an even number of points) then it must be n-extendable. The second result applies to graphs with toughness less than one and presents an upper bound on the value of n for which such a graph can be n-extendable. In the final section, we compare and contrast these results with the n-factor results of Enomoto, Jackson, Katerinis and A. Saito
Matching Extension and Connectivity in Graphs. 1. Introduction and Terminology( Book )

1 edition published in 1986 in English and held by 1 WorldCat member library worldwide

Matching Extension in Regular Graphs( Book )

1 edition published in 1989 in English and held by 1 WorldCat member library worldwide

This paper deals with extending matching in regular graphs. There are two main results. The first presents a sufficient condition in terms of cyclic connectivity for extending matching in regular bipartite graphs. This theorem generalizes an earlier result due to Holton and the author. The second result deals with regular-but not necessarily bipartite-graphs. In this case, it is known that a result analogous to that obtained in the bipartite case is impossible, but a new proof is given of a result of Naddef and Pulleyblank which guarantees that a regular graph with an even number of points which has sufficiently large cyclic connectivity will be bicritical. Algorithms. (jes)
Collocation at Gauss Points as a Discretization in Optimal Control( Book )

1 edition published in 1977 in English and held by 1 WorldCat member library worldwide

Matching Extension and the Genus of a Graph( Book )

1 edition published in 1986 in English and held by 1 WorldCat member library worldwide

Let G be a graph with p points having a perfect matching and suppose n is a positive integer with n <or = (p-2)/2. Then G is n-extendable if every matching in G containing n lines is a subset of a perfect matching. In this paper we obtain an upper bound on the n-extendability of a graph in terms of its genus. Keywords: Euler contributions; Theorems. (Author)
On the 2-Extendability of Planar Graphs( Book )

1 edition published in 1989 in English and held by 1 WorldCat member library worldwide

Some sufficient conditions for the 2-extendability of k-connected k-regular (k> or = 3) planar graphs are given. In particular, it is proved that for k> or = 3, a k-connected k-regular planar graph with each cyclic cutset of sufficiently large size is 2-extendable. All graphs in this paper are finite, undirected, connected and simple, although some parallel edge situations will occur after some contractions are made. However, any loops formed by these contractions will be deleted. Let nu and n be positive integers with n <or = (v - 2)/2 and let G be a graph with nu vertices and epsilon edges having a perfect matching. The graph G is said to be n-extendable if every matching of size n in G lies in a perfect matching of G
On the Cyclability of k-Connected (k+1)-Regular Graphs( Book )

1 edition published in 1986 in English and held by 1 WorldCat member library worldwide

In the past fifteen years or so, there have been quite a number of papers dealing with variations on the following general theme. Given a graph G and a positive integer m, m <or = /V(G), find non-trivial conditions on G which will guarantee that given a set S = (v sub 1 ..., v sub m) - V(G), there exists a cycle C sub S containing S. In the special case m = /V(G), this documents deals with conditions for the existence of Hamiltonian cycles, in itself a subject studied extensively by many graph theorists
Some Recent Results on Graph Matching( Book )

1 edition published in 1987 in English and held by 1 WorldCat member library worldwide

A matching in a graph G is a set of lines, no two of which share a common point. A matching is perfect if it spans V(G). The problem of finding a matching of maximum cardinality in a graph models a number of significant real-world problems and, in addition, is of considerable mathematical interest in its own right. Matchings are in a sense among the best understood graph-theoretic objects: there exist efficient algorithms to find and good characterizations for the existence of perfect matchings and for the maximum weight of a matching; there are nice descriptions of polyhedra associated with matchings; good bounds and, for a few special classes, exact formulas for the number of perfect matchings in a graph. But there are many important questions that remain unanswered. What is the number of perfect matchings in a general graph? Which graphs can be written as the disjoint union of perfect matchings (i.e., which r-regular graphs are r-line-colorable)? How does one generate a random perfect matching? Matching theory has often been in the front lines of research in graph theory and many results in matching theory have served as pilot results for new branches of study in combinatorics (e.g., minimax theorems, good characterizations and polyhedral descriptions)
Claw-Free Maximal Planar Graphs( Book )

1 edition published in 1989 in English and held by 1 WorldCat member library worldwide

A graph G is claw-free if it contains no induced subgraph isomorphic to the complete bipartite graph K. Such graphs have been widely studied with respect to such other graph properties as matching, perfection, vertex-packing, Hamiltonian cycles and related questions on traversability, and reconstruction. A planar graph is said to be maximal planar (or a triangulation) if, given any imbedding of G in the plane, every face boundary is a triangle. The abbreviations MAXP and CFMAXP are used for the properties maximal planar and claw-free maximal planar respectively. (Recall that every maximal planar graph with at least three points is either the complete graph K3 or else is 3- connected and thus it follows that such a graph has a unique imbedding in the plane)
Homogeneously Traceable Results in Claw-Free Graphs( )

1 edition published in 1993 in English and held by 0 WorldCat member libraries worldwide

A graph is homogeneously traceable if for each v in V(C) there is a Hamilton path starting at v. In this paper we find a sufficient condition for a claw-free graph to be homogeneously traceable in terms of a neighborhood union condition. Hamilton path, Neighborhood union
Analysis of Bivariate and Trivariate Macro-Elements for Surface Fitting( )

1 edition published in 2002 in English and held by 0 WorldCat member libraries worldwide

The work performed under this grant has focused on developing specific spline tools (and the necessary underlying theory). The results can be of use for a variety of applied problems, including for example a) scattered data fitting of very large data sets (such as arise in digital terrain modelling, geosciences, meteorology, etc.), and b) numerical solution of boundary-value problems by finite-element methods. The work has resulted in a total of 16 research papers
A Class of Planar Well-covered Graphs With Girth Four( )

1 edition published in 1991 in English and held by 0 WorldCat member libraries worldwide

A well-covered graph is a graph in which every maximal independent set is a maximum independent set; Plummer introduced the concept in a 1970 paper. The notion of a 1 -well- covered graph was introduced by Staples in her 1975 dissertation: a well-covered -graph G is 1 -well-covered if and only if G-v is also well-covered for every point v in G. Except for K.) and C5, every 1- well-covered graph contains triangles or 4-cycles. Thus, triangle-free 1 -well- covered graphs necessarily have girth 4. We show that all planar 1-well-covered graphs of girth 4 belong to a specific infinite family, and we give a characterization of this family
Extending Matchings in Planar Graphs 4( )

1 edition published in 1989 in English and held by 0 WorldCat member libraries worldwide

The structure of certain non-2-extendable planar graphs is studied first. In particular, 4-connected 5-regular planar graphs which are not 2-extendable are investigated and examples of these are presented. It is then proved that all 5-connected even planar graphs are 2-extendable. Finally, a certain configuration called a generalized butterfly is defined and it is shown that 4-connected maximal planar even graphs which contain no generalized butterfly are 2-extendable
The Cartesian Product of a k-Extendable and an l-Extendable Graph is (k + l +1)-Extendable( )

1 edition published in 1991 in English and held by 0 WorldCat member libraries worldwide

Let us start with the definition of a kappa-extendable graph G. Suppose kappa is an integer such that 1 <or = kappa <or = (/V(G)/-2)/2. A graph G is kappa-extendable if G is connected, has a perfect matching (a 1- factor) and any matching in G consisting of kappa edges can be extended to (i.e., is a subset of) a perfect matching. The extendability number of G, extG, is the maximum kappa such that G is kappa-extendable. A natural problem is to determine the extendability number of a graph G
Matching and Vertex Packing: How Hard Are They?( )

1 edition published in 1991 in English and held by 0 WorldCat member libraries worldwide

Two of the most well-known problems in graph theory are: Find a maximum matching (or perfect matching, if one exists); and Find a maximum independent set of vertices. The first problem-usually called the matching problem-is known to have a polynomial algorithm; the second-often called the vertex packing problem-is known to be NP-complete. However, many graph theorists-especially those who do not deal much with complexity of algorithms- know little more about the complexity issues associated with these two problems than these two basic facts. What is not so widely known within the graph theory community is that these two problems have motivated a great deal of recent activity in the area of algorithms and their complexity. Of course it is not known whether or not P = NP, but most workers in the area currently believe that equality is unlikely to hold. Motivated by this belief, a number of people have studied variations of both matching and vertex packing with the general theme being two-fold. On the one hand, one can add various side conditions to the matching problem and study the complexity-both sequential and parallel-of the resulting problems. On the other hand, one can investigate certain large and interesting classes of graphs trying to prove that for these classes the vertex packing problem has a polynomial solution
Well-Covered Graphs: A Survey( )

1 edition published in 1991 in English and held by 0 WorldCat member libraries worldwide

A graph G is well-covered (or w-c) if every maximal independent set of points in G is also maximum. Clearly, this is equivalent to the property that the greedy algorithm for constructing a maximal independent set always results in a maximum independent set. Although the problem of independence number is well-known to be NP-complete, it is trivially polynomial for well covered graphs. The concept of well-coveredness was introduced by the author in PI and was first discussed therein with respect to its relationship to a number of other properties involving the independence number. Since then, a number of results about well-covered graphs have been obtained. It is our purpose in this paper to survey these results for the first time. As the reader will see, many of the results we will discuss are quite recent and have not as yet appeared in print
Matching Theory - A Sampler: From Denes Koenig to the Present( )

1 edition published in 1991 in English and held by 0 WorldCat member libraries worldwide

It was Koenig who gave the next strong impetus to the study of graph factorization after Petersen's ground breaking work, and it is Koenig with whom we are charged to begin out brief summary of the history of matching theory. Fortunately, matching theory serves well as an historical thread extending from the time of Koenig (and before) up to the present, wending its way through graph theory and intersecting many of the most important new ideas which have sprung forth in our discipline. One sees in particular that after the close of World War II this intertwining of matching theory with the study of graphs as a whole became ever more inextricable, even as the study of graph theory as a discipline unto itself literally exploded upon the mathematical scene. Although it is jumping the gun somewhat with respect to the organization of this paper, we can mention three major areas which have joined with graph theory to give rise to many new and deep results. These are: (1) linear programming and polyhydral combinatorics; (2) the linking of graphs and probability theory in the area of random graphs and finally (3) the theory of algorithms and computational complexity
Extending Matchings in Graphs: A Survey( )

1 edition published in 1990 in English and held by 0 WorldCat member libraries worldwide

This paper surveys a variety of results obtained over the past few years concerning n-extendable graphs. In particular, we will describe how the property of n-extendability interacts with such other graph parameters as genus, toughness, claw-freedom and degree sums and generalized neighborhood conditions. We will also investigate the behavior of matching extendability under the operation of Cartesian product. The study of n-extendability for planar graphs has been-and continues to be-of particular interest
A Class of Well-Covered Graphs With Girth Four( )

1 edition published in 1991 in English and held by 0 WorldCat member libraries worldwide

A graph is well-covered if every maximal independent set is also a maximum independent set. A 1-well-covered graph G has the additional property that G-v is also well-covered for every point v in G. Thus, the 1-well-covered graphs form a subclass of the well-covered graphs. We examine triangle-free 1- well-covered graphs. Other than C5 and K2, a 1-well-covered graph must contain a triangle or a 4-cycle. Thus, the graphs we consider have girth 4. Two constructions are given which yield infinite families of 1-well-covered graphs with girth 4. These families contain graphs with arbitrarily large independence number
2-Extendability in Two Classes of Claw-Free Graphs( )

1 edition published in 1992 in English and held by 0 WorldCat member libraries worldwide

A graph G is 2-extendable if it has at least six vertices and every pair of independent edges extends to (i.e., is a subset of) a perfect matching. In this paper two classes of claw-free graphs are discussed: those which are 3- regular and 3-connected and those which are 4-regular and 4-connected (as well as even). None of the first class is 2-extendable, whereas those of the second class which are 2-extendable are determined. More particularly in the graphs belonging to these classes, those pairs of independent edges which extend to a perfect matching are determined

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