VANDERBILT UNIV NASHVILLE TN Dept. of MATHEMATICS
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Works:  21 works in 22 publications in 1 language and 22 library holdings 

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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 nextendability 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 nextendable. 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 nextendable. In the final section, we compare and contrast these results with the nfactor results of Enomoto, Jackson, Katerinis and A. Saito
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 nextendability 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 nextendable. 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 nextendable. In the final section, we compare and contrast these results with the nfactor 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
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 regularbut not necessarily bipartitegraphs. 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)
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 regularbut not necessarily bipartitegraphs. 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
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 = (p2)/2. Then G is nextendable 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 nextendability of a graph in terms of its genus. Keywords: Euler contributions; Theorems. (Author)
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 = (p2)/2. Then G is nextendable 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 nextendability of a graph in terms of its genus. Keywords: Euler contributions; Theorems. (Author)
On the 2Extendability of Planar Graphs(
Book
)
1 edition published in 1989 in English and held by 1 WorldCat member library worldwide
Some sufficient conditions for the 2extendability of kconnected kregular (k> or = 3) planar graphs are given. In particular, it is proved that for k> or = 3, a kconnected kregular planar graph with each cyclic cutset of sufficiently large size is 2extendable. 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 nextendable if every matching of size n in G lies in a perfect matching of G
1 edition published in 1989 in English and held by 1 WorldCat member library worldwide
Some sufficient conditions for the 2extendability of kconnected kregular (k> or = 3) planar graphs are given. In particular, it is proved that for k> or = 3, a kconnected kregular planar graph with each cyclic cutset of sufficiently large size is 2extendable. 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 nextendable if every matching of size n in G lies in a perfect matching of G
On the Cyclability of kConnected (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 nontrivial 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
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 nontrivial 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 realworld problems and, in addition, is of considerable mathematical interest in its own right. Matchings are in a sense among the best understood graphtheoretic 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 rregular graphs are rlinecolorable)? 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)
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 realworld problems and, in addition, is of considerable mathematical interest in its own right. Matchings are in a sense among the best understood graphtheoretic 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 rregular graphs are rlinecolorable)? 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)
ClawFree Maximal Planar Graphs(
Book
)
1 edition published in 1989 in English and held by 1 WorldCat member library worldwide
A graph G is clawfree 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, vertexpacking, 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 clawfree 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)
1 edition published in 1989 in English and held by 1 WorldCat member library worldwide
A graph G is clawfree 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, vertexpacking, 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 clawfree 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 ClawFree 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 clawfree graph to be homogeneously traceable in terms of a neighborhood union condition. Hamilton path, Neighborhood union
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 clawfree graph to be homogeneously traceable in terms of a neighborhood union condition. Hamilton path, Neighborhood union
Analysis of Bivariate and Trivariate MacroElements 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 boundaryvalue problems by finiteelement methods. The work has resulted in a total of 16 research papers
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 boundaryvalue problems by finiteelement methods. The work has resulted in a total of 16 research papers
A Class of Planar Wellcovered Graphs With Girth Four(
)
1 edition published in 1991 in English and held by 0 WorldCat member libraries worldwide
A wellcovered 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 wellcovered graph G is 1 wellcovered if and only if Gv is also wellcovered for every point v in G. Except for K.) and C5, every 1 wellcovered graph contains triangles or 4cycles. Thus, trianglefree 1 well covered graphs necessarily have girth 4. We show that all planar 1wellcovered graphs of girth 4 belong to a specific infinite family, and we give a characterization of this family
1 edition published in 1991 in English and held by 0 WorldCat member libraries worldwide
A wellcovered 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 wellcovered graph G is 1 wellcovered if and only if Gv is also wellcovered for every point v in G. Except for K.) and C5, every 1 wellcovered graph contains triangles or 4cycles. Thus, trianglefree 1 well covered graphs necessarily have girth 4. We show that all planar 1wellcovered 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 non2extendable planar graphs is studied first. In particular, 4connected 5regular planar graphs which are not 2extendable are investigated and examples of these are presented. It is then proved that all 5connected even planar graphs are 2extendable. Finally, a certain configuration called a generalized butterfly is defined and it is shown that 4connected maximal planar even graphs which contain no generalized butterfly are 2extendable
1 edition published in 1989 in English and held by 0 WorldCat member libraries worldwide
The structure of certain non2extendable planar graphs is studied first. In particular, 4connected 5regular planar graphs which are not 2extendable are investigated and examples of these are presented. It is then proved that all 5connected even planar graphs are 2extendable. Finally, a certain configuration called a generalized butterfly is defined and it is shown that 4connected maximal planar even graphs which contain no generalized butterfly are 2extendable
The Cartesian Product of a kExtendable and an lExtendable 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 kappaextendable graph G. Suppose kappa is an integer such that 1 <or = kappa <or = (/V(G)/2)/2. A graph G is kappaextendable 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 kappaextendable. A natural problem is to determine the extendability number of a graph G
1 edition published in 1991 in English and held by 0 WorldCat member libraries worldwide
Let us start with the definition of a kappaextendable graph G. Suppose kappa is an integer such that 1 <or = kappa <or = (/V(G)/2)/2. A graph G is kappaextendable 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 kappaextendable. 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 wellknown 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 problemusually called the matching problemis known to have a polynomial algorithm; the secondoften called the vertex packing problemis known to be NPcomplete. However, many graph theoristsespecially 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 twofold. On the one hand, one can add various side conditions to the matching problem and study the complexityboth sequential and parallelof 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
1 edition published in 1991 in English and held by 0 WorldCat member libraries worldwide
Two of the most wellknown 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 problemusually called the matching problemis known to have a polynomial algorithm; the secondoften called the vertex packing problemis known to be NPcomplete. However, many graph theoristsespecially 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 twofold. On the one hand, one can add various side conditions to the matching problem and study the complexityboth sequential and parallelof 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
WellCovered Graphs: A Survey(
)
1 edition published in 1991 in English and held by 0 WorldCat member libraries worldwide
A graph G is wellcovered (or wc) 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 wellknown to be NPcomplete, it is trivially polynomial for well covered graphs. The concept of wellcoveredness 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 wellcovered 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
1 edition published in 1991 in English and held by 0 WorldCat member libraries worldwide
A graph G is wellcovered (or wc) 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 wellknown to be NPcomplete, it is trivially polynomial for well covered graphs. The concept of wellcoveredness 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 wellcovered 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
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 nextendable graphs. In particular, we will describe how the property of nextendability interacts with such other graph parameters as genus, toughness, clawfreedom 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 nextendability for planar graphs has beenand continues to beof particular interest
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 nextendable graphs. In particular, we will describe how the property of nextendability interacts with such other graph parameters as genus, toughness, clawfreedom 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 nextendability for planar graphs has beenand continues to beof particular interest
A Class of WellCovered Graphs With Girth Four(
)
1 edition published in 1991 in English and held by 0 WorldCat member libraries worldwide
A graph is wellcovered if every maximal independent set is also a maximum independent set. A 1wellcovered graph G has the additional property that Gv is also wellcovered for every point v in G. Thus, the 1wellcovered graphs form a subclass of the wellcovered graphs. We examine trianglefree 1 wellcovered graphs. Other than C5 and K2, a 1wellcovered graph must contain a triangle or a 4cycle. Thus, the graphs we consider have girth 4. Two constructions are given which yield infinite families of 1wellcovered graphs with girth 4. These families contain graphs with arbitrarily large independence number
1 edition published in 1991 in English and held by 0 WorldCat member libraries worldwide
A graph is wellcovered if every maximal independent set is also a maximum independent set. A 1wellcovered graph G has the additional property that Gv is also wellcovered for every point v in G. Thus, the 1wellcovered graphs form a subclass of the wellcovered graphs. We examine trianglefree 1 wellcovered graphs. Other than C5 and K2, a 1wellcovered graph must contain a triangle or a 4cycle. Thus, the graphs we consider have girth 4. Two constructions are given which yield infinite families of 1wellcovered graphs with girth 4. These families contain graphs with arbitrarily large independence number
2Extendability in Two Classes of ClawFree Graphs(
)
1 edition published in 1992 in English and held by 0 WorldCat member libraries worldwide
A graph G is 2extendable 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 clawfree graphs are discussed: those which are 3 regular and 3connected and those which are 4regular and 4connected (as well as even). None of the first class is 2extendable, whereas those of the second class which are 2extendable are determined. More particularly in the graphs belonging to these classes, those pairs of independent edges which extend to a perfect matching are determined
1 edition published in 1992 in English and held by 0 WorldCat member libraries worldwide
A graph G is 2extendable 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 clawfree graphs are discussed: those which are 3 regular and 3connected and those which are 4regular and 4connected (as well as even). None of the first class is 2extendable, whereas those of the second class which are 2extendable 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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