Showing papers on "Complement graph published in 1973"

Dividing a Graph into Triconnected Components

Abstract: An algorithm for dividing a graph into triconnected components is presented. When implemented on a random access computer, the algorithm requires $O(V + E)$ time and space to analyze a graph with V vertices and E edges. The algorithm is both theoretically optimal to within a constant factor and efficient in practice.

Testing flow graph reducibility

Abstract: Many problems in program optimization have been solved by applying a technique called interval analysis to the flow graph of the program. A flow graph which is susceptible to this type of analysis is called reducible. This paper describes an algorithm for testing whether a flow graph is reducible. The algorithm uses depth-first search to reveal the structure of the flow graph and a good method for computing disjoint set unions to determine reducibility from the search information. When the algorithm is implemented on a random access computer, it requires O(E log* E) time to analyze a graph with E edges, where log* x = min{i/logix≤1}. The time bound compares favorably with the O(E log E) bound of a previously known algorithm.

Graph theorems for manifolds

Abstract: Two basic theorems about the graphs of convex polytopes are that the graph of ad-polytope isd-connected and that it contains a refinement of the complete graph ond+1 vertices. We obtain generalizations of these theorems, and others, for manifolds. We also supply some details for a proof of the lower bound inequality for manifolds.

On the Perfect Graph Theorem

Abstract: Publisher Summary This chapter provides an overview of perfect graph theorem. A version of the perfect graph theorem says, “Let A be a (0, l)-matrix such that the linear program yA w, y 0, min 1. y (where 1 = (1,. . . , 1)) always has an integer solution vector y whenever w is a (0, 1)-vector, then this program always has an integer solution vector y whenever w is a non-negative integer vector. The chapter discusses some well-known integer-valued functions of an arbitrary graph. Another perfect graph theorem says that if G is γ-perfect (or π-perfect), then G is perfect. A stronger form, one that is still open, asserts that G is perfect if and only if neither G nor its complement G contains an odd hole. The pluperfect graph theorem is that if G is γ-pluperfect (or π-pluperfect), then G is pluperfect. Thus, to prove the perfect graph theorem, it would suffice to show that if G is π-perfect, then G is also π-pluperfect.

On a New Approach for Finding All the Modified Cut-Sets in an Incompatibility Graph

Abstract: The compatibility relation occurs in many different disciplines in science and engineering. When a compatibility relation exists between pairs of elements in a set, an important problem is to derive the collection of aU those elements that form maximal compatibles. If the set of elements with the compatibility relation can be visualized as a compatibility graph of which the different nodes represent the elements of the set, the only edges of the graph being the nonoriented lines joining pairs of elements with the compatibility relation, then the problem of deriving the maximal compatibles becomes identical to the graph theory problem of finding all the maximal complete subgraphs in a symmetric graph. Recently, in connection with simplifying incompletely specified sequential machines, where a kind of compatibility relation also exists between pairs of internal states, Das and Sheng proposed a method for deriving the different maximal compatibles through finding all of the modified cut-sets of the incompatibility graph of the machine. This paper, without confining itself to only incompletely specified machines, considers the problem involving the compatibility relation in a broader perspective and suggests a new approach for finding aU the modified cut-sets of the incompatibility graph of a set having a compatibility relation between its different pairs of elements.

Two-step encoding for finite sources

Abstract: Any finite information source is given a graph structure, in which two vertices are adjacent whenever the two corresponding source letters are distinguishable by the coder-decoder pair. Usual sources correspond, therefore, to complete graphs. If the associated graph is not complete, however, an \varepsilon -code for the source can be constructed in two steps: in the first, distinct codewords are given to distinguishable letters only; in the second step, a similar encoding is carried out for the complementary graph, in which distinguishable letters become indistinguishable and the converse. A particularly simple case shows up when nonadjacency is an equivalence relation among the vertices of the graph: each class of nondistinguishable letters can then be considered as a letter in a coarser source alphabet. The two-step procedure is then particularly intuitive. A problem arises when this procedure does not destroy optimality of the resulting \varepsilon -code; some partial results are given in this direction. The results obtained are largely based on some graph-theoretical ideas and tools.

The existence of 1-factors in line graphs, squares, and total graphs

Abstract: The line graph L ( G ) of G is that graph whose vertex set corresponds to the edge set of G such that two vertices of L ( G ) are adjacent if and only if the corresponding edges of G are adjacent. The square G 2 of G has the same vertex set as G and two vertices are adjacent if and only if their distance in G is at most two. The total graph T ( G ) of G is the graph whose vertex set corresponds to the set of vertices and edges of G such that two vertices of T ( G ) are adjacent if and only if the corresponding elements of G are adjacent or incident. A 1-factor of a graph is a spanning 1-regular subgraph. For a connected graph G , it is shown that a necessary and sufficient condition that L ( G ) (respectively, G 2 ; respectively, T ( G )) have a 1-factor is that L ( G ) (respectively, G 2 ; respectively, T ( G )) have an even number of vertices.

Graphs with given connectivity and independence number or networks with given measures of vulnerability and survivability

Abstract: The criterion for invulnerability of a network based on the connectivity of a graph is well treated in literature. We define a criterion for survivability of a network in terms of the independence number of a graph. The following problems are then considered. 1) Find an r -connected graph, with n vertices and m_0 = [(nr + l)/2] edges, whose independence number is the minimum possible, where [x] denotes the greatest integer less than or equal to x \cdot 2 ) Given positive integers n,m,r , and k , find the realizability conditions for a graph with n vertices and m edges, whose connectivity is at least r and independence number is at most k .

On Spectrally Bounded Graphs

Abstract: Publisher Summary This chapter discusses spectrally bounded graphs. Two vertices are adjacent if they are joined by an edge. The valence of a vertex is the number of vertices adjacent to it. The set of vertices of G (graph) is denoted by V(G) and the set of edges by E(G). If G is a graph, G is the graph with V(G) = V(G), and two distinct vertices are adjacent in G if and only if they are not adjacent in G. The chapter focuses on large cliques in G and explains an equivalence relation on large cliques. The equivalence classes of large cliques are shown to have several properties and explained that if additional edges are added, the equivalence classes would be cliques.

Uses of Graph Theory

Abstract: Graph Theory and Computing. Edited by Ronald C. Read. Pp. xiv + 329. (Academic: New York and London, October 1972.) $17.50.

Balanced Hypergraphs and Some Applications to Graph Theory

Abstract: Publisher Summary This chapter provides an overview of the combinatorial properties of a special kind of hypergraphs, called the balanced hypergraphs, which provide new theorems of graph theory. A hypergraph consists of a finite set X of n vertices together with a family of m nonempty subsets of X, called the edges. A graph is made up of vertices or nodes and lines called edges that connect them. A graph can be undirected, that is, there is no distinction between the two vertices associated with each edge, or its edges can be directed from one vertex to another; Graphs are one of the prime objects of study in discrete mathematics.