Abstract: Graph augmentation problems on a weighted graph involve determining a minimum-cost set of edges to add to a graph to satisfy a specified property, such as biconnectivity, bridge-connectivity or strong connectivity. These augmentation problems are shown to be NP-complete in the restricted case of the graph being initially connected. Approximation algorithms with favorable time complexity are presented and shown to have constant worst-case performance ratios.

Abstract: A random graph with (1+e)n/2 edges contains a path of lengthcn. A random directed graph with (1+e)n edges contains a directed path of lengthcn. This settles a conjecture of Erdos.

Abstract: Finding the solution of a dynamic programming problem m the form of polyadlc funcUonal equatmns is shown to be equivalent to searching a mmmaal cost path in an AND/OR graph with monotone cost functions The proof is given in an algebraic framework and is based on a commutaUvity result between solutton and mterpretauon of a symbohc system This approach Is simdar to the one used by some authors to prove the eqmvalence between the operaUonal and denotatmnal semantics of programming languages

Abstract: Lemma 3 of [1] makes a false assertion about the winding number w(P, IT). A counterexample to the lemma is obtained by choosing Px = (123), P2 = (654), and n = {{1, 4}, {2, 5}, {3, 6}}. Here w(P,, n) = <¿iP2, IT) = 1, whereas w(P,P2, IT) = 0. This also invalidates the proof of the lower bound of Theorem 22 of [1] on the genus of the amalgamation of graphs over three points. More details will appear in [2].

Abstract: Structures of graph theory are compared with those of Q-analysis and there are many similarities. The graph and simplicial complex defined by a relation are equivalent in terms of the information they represent, so that the choice between graph theory and Q-analysis depends on which gives the most natural and complete description of a system. The higher dimensional graphs are shown to be simplicial families or complexes. Although network theory is very successful in those physical science applications for which it was developed, it is argued that Q-analysis gives a better description of human network systems as patterns of traffic on a backcloth of simplicial complexes. The q-nearness graph represents the q-nearness of pairs of simplices for a given q-value. It is concluded that known results from graph theory could be applied to the q-nearness graph to assist in the investigation of q-connectivity, to introduce the notion of connection defined by graph cuts, and to assist in computation. The application ...

Abstract: While the concept of the graph center is unambiguous (and quite old) in the case of acyclic graphs, an attempt has been made recently to extend the concept to polycyclic structures using the distance matrix of a graph as the basis. In this work we continue exploring such generalizations considering in addition to the distance matrix, self-avoiding walks or paths as graph invariants of potential interest for discriminating distinctive vertex environments in a graph of polycyclic structures. A hierachy of criteria is suggested that offers a systematic approach to the vertex discrimination and eventually establishes in most cases the graph center as a single vertex, a single bond (edge), or a single group of equivalent vertices. Some applications and the significance of the concept of the graph center are presented.

Abstract: Directed node-label controlled graph grammars (DNLC grammars) are sequential graph rewriting systems. In a direct derivation step of a DNLC grammar a single node is rewritten. Both the rewriting of a node and the embedding of a "daughter graph" in a "host graph" are controlled by the labels of nodes only. We study the use of those grammars to define string languages. In particular we provide a characterization of the class of context-free string languages in terms of DNLC grammars.

Abstract: We express the Quadratic Assignment Problem in terms of graph multiplication of a flow graph G f with a distance graph G d . By decomposing G f into simpler graphs, we give a general unified procedure to calculate bounds. This procedure generates the two previously known bounds as special cases and also generates some new bounds. By enumerating all practical decompositions, we show that no better bounds can be computed in reasonable time—by solving linear assignment subproblems—other than the bounds pointed out in this paper.

Abstract: We show that the edges of the complete symmetric directed graph onn vertices can be partitioned into directed cycles (or anti-directed cycles) of lengthn−1 so that any two distinct cycles have exactly one oppositely directed edge in common whenn=p e>3, wherep is a prime ande is a positive integer. When the cycles are anti-directedp must be odd. We then consider the designs which arise from these partitions and investigate their construction.

Abstract: This correspondence generalizes Hayes' recent ideas for generating an optimal transition write sequence which forms the "backbone" of his algorithm for testing semiconductor RAM's for pattern-sensitive faults. The generalization, presented in graph theoretic terms, involves two sequential steps. The frmst step results in assigning of a "color" to each memory cell. In the second step, each color is defined as a distinct sequence of bits representing the sequence of states assumed by the correspondingly colored cell. The constraints imposed at each step lead to interesting and general problems in graph theory: the standard graph coloring problem in the first step, and a path projection problem from a binary m-cube to a subcube in the second step. Applications to arbitrary k-cell neighborhoods, and particularly to three-cell neighborhoods are shown.

Abstract: PART I 1. BASIC DEFINITIONS AND RESULTS Using the classical Closed Graph Theorem for Banach spaces as a starting point, several different characterizations of a function having a closed graph are provided. The concept of a "cluster set" of a function at a point is used to simplify the proofs of some well known theorems. Here some of the standard results are included, such as the ones in 12 and 43 . 2. THE CLOSED GRAPH AND MINIMAL TOPOLOGICAL SPACES There are several generalizations of compact and compact Hausdorff spaces of current interest. These spaces are generally referred to as minimal topological spaces. In section I.l, compactness plays a very important role. In this section we show how the closed graph property interacts with these more general spaces. 3. SOME APPLICATIONS TO FUNCTIONAL ANALYSIS The Closed Graph Theorem is a very powerful tool in Banach spaces. Our investigation of the closed graph property in the more general setting of functions in topological spaces of course leads to sufficient conditions for and characterizations of the closed graph property, which have implications in Banach spaces. In particular, the Closed Graph Theorem is rephrased in a new and interesting way. 4. NON-CONTINUOUS FUNCTIONS AND THE CLOSED GRAPH PROPERTY Functions with closed graphs are but one of many types of non-continuous functions. In this section the relationship between the closed graph property and some of these functions is explored. This investigation leads to several results which give sufficient conditions for continuity. 5. CHARACTERIZATIONS OF COMPACTNESS AND COUNTABLE COMPACTNESS Th~re is an interesting relationship between the concepts of compactness and the closed graph property. It is shown how the closed graph property can be used to characterize compactness and countable compactness where the space in question is used as a domain or a range space. 6. POINTS OF DISCONTINUITY Since functions that have closed graphs and continuous functions have many similar properties, the set of discontinuities of a function with a closed graph should, and does, lend itself to investigation. The Baire space property, perfect normality, and metric spaces with the property that bounded sets have compact closure, all play an important role in describing this set as closed and nowhere dense under appropriate conditions.

Abstract: An embedding of the graph G in the graph H is a one-to-one association of the vertices of G with the vertices of H. There are two natural measures of the cost of a graph embedding, namely, the dilation-cost of the embedding: the maximum distance in H between the images of vertices that are adjacent in G, and the expansion-cost of the embedding: the ratio of the size of H to the size of G. The main results of this paper illustrate three situations wherein one of these costs can be minimized only at the expense of a dramatic increase in the other cost. The first result establishes the following: there is an embedding of n-node complete ternary trees in complete binary trees with dilation-cost 2 and expansion-cost Θ(nλ) where λ=log3(4/3); but any embedding of these ternary trees in binary trees that has expansion-cost <2 must, infinitely often, have dilation-cost ⩾ (const)log log log n. The second result provides a stronger but less easily stated example of the same type of tradeoff. The third result concerns generic binary trees, that is, complete binary trees into which all n-node binary trees are "efficiently" embeddable. There is a generic binary tree into which all n-node binary trees are embeddable with dilation-cost O(1) and expansion-cost O(nc) for some fixed constant c; if one insists on embeddings whose dilation-cost is exactly 1, then these embeddings must have expansion-cost ω(n(log n)/2); if one insists on embeddings whose expansion-cost is <2, then these embeddings must, infinitely often, have dilation-cost ⩾ (const)log log log n. An interesting application of the polynomial size generic binary tree in the first part of this three-part result is to yield simplified proofs of several results concerning computational systems with an intrinsic notion of "computation tree", such as alternating and nondeterministic Turing machines and context-free grammars.

Abstract: A graph product is the fundamental group of a graph of groups. Amongst the simplest examples are HNN groups and free products with amalga- mation. The torsion and conjugacy theorems are proved for any group presented as a graph product. The graphs over which some graph product has a solvable word problem are characterised. Conditions are then given for the solvability of the word and order problems and also for the extended word problem for cyclic subgroups of any graph product. These results generalise the known ones for HNN groups and free products with amalgamation. 1. Introduction. Interest in groups which are graph products (that is, fundamental groups of graphs of groups in the terminology of Bass and Serre) has been marked in recent years, in view of their utility in combinatorial group theory. The normal form theorem (Britton's Lemma) and the conjugacy theorem (Collins' Lemma) for the simplest graph products-the HNN groups and free products with amalgama- tion-have been vital in the study of decision problems for these groups. This paper proves the torsion and conjugacy theorems for graph products. Decision problems for a graph product cannot be considered without first paying attention to the associated graph. The graphs over which some graph product has a solvable word problem are completely characterised. (Any such solution is not presentation-free: a group may have two presentations as a graph product over distinct graphs, of which only one has a solvable word problem.) Conditions are given for the solvability of the word and order problems, and also for the solvability of the extended word problem for cyclic subgroups of graph products. The known results for HNN groups, free products with amalgamation and their various generalisations follow immediately as particular cases. A class of graph products is defined for which the word problem and extended word problem for infinite cyclic subgroups are both solvable. The work is couched in terms of (Brandt) groupoids, which provide a most natural setting for proofs of this kind. Hence a graph product is considered as the loop group of a connected mapping cylinder of a diagram of groups, in which all the group homomorphisms are monic. The reader is referred to (3) for the basic constructions of the theory of groupoids, to (8) for the Bass-Serre theory, and to (6, Chapter V) for the theory of computability.

Abstract: Connection graph resolution is a refinement of resolution introduced by Kowalski in the early seventies /2/. However, it is still an open problem whether this proof rule (like resolution) is complete in the strong sense that for any unsatisfiable formula any sequence of selections of connections to be resolved upon leads to a refutation provided each connection has a finite chance to be selected.

Abstract: A parallel connection graph proof procedure is described which is going to be implemented in CSSA, a new language for asynchronous multiprocessor systems

Abstract: Determining whether or not a directed graph has a kernel belongs to a class of hard combinatorial problems, known as NP-complete. In this paper we develop a backtracking algorithm which generates all the kernels of a directed, graph in lexicographic.order. Extensive computational experience on randomly generated graphs ranging from 10 to 100 nodes and from 30% to 90% densities has shown that this algorithm compares very favourably with existing algorithms.

Abstract: The graph of O'Keefe and Wong, with valency 7, girth 6, and 90 vertices, is constructed as a 3-fold covering graph, and it is shown that there is a unique covering graph with these properties.

Abstract: The thesis consists of three chapters. The first chapter introduces the basic notions of graph theory and defines vertex-reconstruction and edge-reconstruction problem. The second chapter and third chapter are devoted to the edge-reconstruction of bi-degreed graphs and bipartite graphs respectively. A bi-degreed graph G is a graph with two degrees d > δ. By elementary arguments we can assume d = δ + 1 and there are at least two vertices of degree δ. Call vertices of degree d "big" vertex and degree δ "small" vertex. Define "symmetric" path of length p Sp to be one with both ends small vertices and all other internal vertices big vertices; define "asymmetric" path of length p Ap to be one with one end a small vertex and all others big vertices. If s(G) is the minimum distance between two small vertices in G, we can show that s(G) is "independent" of G (i.e. it is edge-reconstructable), and that G has at most one nonisomorphic edge-reconstruction H. From this, the concept of "forced move" posed by Dr. Swart is obvious. Using the principle of forced move (and sometimes also "forced edge" posed by Dr. Swart as well), it's easy to derive a few interesting properties, like say G is edge-reconstructable if s(G) is even or if two Ss(G)'s intersect at an internal vertex, etc. Write s for s(G). When s is odd, consider the concept of s - n-chain, which is n Ss's following from end to end. We can show first s - 3-chain and then s - 2-chain cannot exist. Hence Ss's are disjoint. Think of Ss's as "lines" in some geometry. Define two more "distance" functions s1 and s2 such that s1 "represents" the distance from a point to a line and s2 means the distance between to "skew" lines. With the aid of forced move principle again, we can at last prove every bi-degreed graph with at least four edges is edge-reconstructable. A bipartite graph G is a graph whose vertex set V is the disjoint union of two sets v1 and v2 such that every edge joins v1 and v2. By elementary reduction we can assume G to be connected. We define special chains inductively so that it starts at a vertex of minimum degree and always goes to a neighbor or minimum degree. Special chains will be the main tool to prove edge-reconstructability. By G's finiteness, we note they will "terminate" somewhere, and we have three types of termination for them. Let condition A•s be that degree sequence of special chain is edge-reconstructable, condition Bi's be that number of special chains is edge-reconstructable(and some more general variations); condition P's be that the "last vertices" of two special chains be not adjacent; we can prove that all A, Bi and P's should hold inductively in an interlocked way. (This is a big task). Then condition P's can be used to prove G's edge-reconstructability for all three types of termination. We can then prove every bipartite graph with at least four edges is edge-reconstructable.

Abstract: : A probabilistic graph consists of vertices and links that fail with some known probabilities. For such a graph, overall reliability is the probability that there exists communication between all vertex-pairs. In this paper, some useful results are presented to simplify the overall reliability computation of an undirected graph when the failure events of the links are statistically independent. (Author)

Abstract: The concept of duality is important in many diverse mathematical areas. In this paper we demonstrate a natural equivalence between dual concepts in two distinct mathematical subjects, linear programming and graph theory.