Showing papers on "Complement graph published in 1977"

A New Algorithm for Generating All the Maximal Independent Sets

Abstract: The problem of generating all the maximal independent sets (or maximal cliques) of a given graph is fundamental in graph theory and is also one of the most important in terms of the application of graph theory. In this paper, we present a new efficient algorithm for generating all the maximal independent sets, for which processing time and memory space are bounded by $O(nm\mu)$ and $O(n+m)$, respectively, where n, m, and $\mu$ are the numbers of vertices, edges, and maximal independent sets of a graph.

On the distance matrix of a directed graph

Abstract: In this note, we show how the determinant of the distance matrix D(G) of a weighted, directed graph G can be explicitly expressed in terms of the corresponding determinants for the (strong) blocks Gi of G. In particular, when cof D(G), the sum of the cofactors of D(G), does not vanish, we have the very attractive formula .

The spanning subgraphs of eulerian graphs

Abstract: It is shown that a connected graph G spans an eulerian graph if and only if G is not spanned by an odd complete bigraph K(2m + 1, 2n + 1). A disconnected graph spans an eulerian graph if and only if it is not the union of the trivial graph with a complete graph of odd order. Exact formulas are obtained for the number of lines which must be added to such graphs in order to get eulerian graphs.

Smallest Augmentations to Biconnect a Graph

Abstract: We provide an $O(| V | + | E |)$ algorithm which, given a graph G, finds a smallest set of edges which, when added to G, produces a graph with no cutpoints

The biparticity of a graph

Abstract: The biparticity β(G) of a graph G is the minimum number of bipartite graphs required to cover G. It is proved that for any graph G, β(G) = {log2χ(G)}. In view of the recent announcement of the Four Color Theorem, it follows that the biparticity of every planar graph is 2.

Parallel Algorithms for Graph Theoretic Problems

Abstract: : The existence of parallel computers has motivated the development of parallel problems solving techniques for many problems. Techniques are studied for solving graph problems on an unbounded parallel model of computation. It is shown that solutions to graph problems can be organized to reveal a large amount of parallelism, which can be exploited to substantially reduce the computation time. Precisely, for an appropriate measure of time complexity, algorithms of time complexity 0 log-squared n are developed to solve each of the following problems for graphs with n vertices: finding minimum spanning trees, biconnected components, dominators, bridges, cycles, cycle bases, and shortest cycles. The number of processors needed to execute each algorithm is bounded above by a polynomial function of n. It is shown that 2 log n + c is a lower bound on the time required to solve each of these graph problems. Thus, the algorithms obtained have time complexities which are optimal to within a factor of log n.

On Partitioning Program Graphs

Abstract: In recent years, applications of graph theory to computer software have given fruitful results and attracted more and more attention. A program graph is a graph structural model of a program exhibiting the flow relation or connection among the elements (statements) in the program.

On vector spaces associated with a graph

Abstract: Four topological entities are necessary for a complete description of a network graph, as required by orthogonal network theory: the seg, the circ and two others, introduced in this paper, as complementary to the seg and the circ respectively. A vector space and its dual are associated with a nonoriented graph. Both spaces are decomposed in two complementary subspaces. The vectors of each subspace share the topological properties of one of the four entities singled out. The results obtained are extended to a directed graph by associating two dual Z-modules with it. The given description of the network graph is perfectly functional to the orthogonal network theory.

The spectrum of a graph

Abstract: We survey the results obtained by a large number of authors concerning the spectrum of a graph. The questions of characterisation by spectrum, cospectral graphs and information derived from the spectrum are discussed.

On factors of a graph

Abstract: Let G be a graph with multiple edges. Let f be a function from the vertex set V(G) of G to the non-negative integers. An f-factor of G is a spanning subgraph F of G such that the degree (valence) of each vertex x in F is f(x). A theorem of Fulkerson, Hoffman and McAndrew [1] gives necessary and sufficient conditions to have an f-factor for a graph G with the odd-cycle property; i.e., if G has the property that either any two of its odd (simple) cycles have a common vertex, or there exists a pair of vertices, one from each cycle, which is joined by an edge. They proved this theorem using integer programming techniques, with a rather long proof. We show that this is a corollary of Tutte's f-factor theorem.

On the relation between graph grammars and graph L-systems

Abstract: Sequential graph rewriting systems called graph or web grammars have been extensively studied within the last years, while parallel graph rewriting systems, named graph L-systems, are a very recent topic of research. In the following a relation between two representatives of these graph rewriting models is given: [CF]=[PEGL], where [CF] is the class of context free graph languages in [12], and [PEGL] is the class of propagating extended graph L-languages given in [13]. The validity of this relation in the graph case, opposite to the string case, is due to the fact that graphs derived in both rewriting systems have nonterminal edges. These edges may occur in the derivation of a graph belonging to the language of such a system, but not in the graph itself. Furthermore, the sequential mechanism is very powerful with respect to the transformation of the embeddings of the replaced graphs, and the parallel mechanism is very general with respect to the transformation of the connections from mother nodes to daughter graphs. These tools allow to simulate a sequential derivation step by two parallel ones and, vice versa, to simulate a parallel derivation step by a sequence of sequential steps.

A note on "Some simplified NP-complete graph problems"

Abstract: have made some important observations about the degree-bounds on digraphs for the Hamiltonian cycle problem to be NP. complete. In this note, we show that their degree-bounds can be modified to yield the strongest degree-bounds. The directed Hamiltonian circuit problem with node-degree bounded by 3-out, 3-in has been shown to be NP. complete [I]. This implies that the directed Hamiltonian circuit with node-degree bounded by 3-out, 1-in or 1-out, 3-in is NP-complete by the following construction~ Expand each node by Further they [i] have shown that the largest degree-bounds for which they know the Hamiltonian circuit problem to be easy were 2-out, 1-in or 2-in, 1-out. This largest degree-bounds can be further increased into 2-in, 2-out by a construction similar to that shown above. Hence it follows that the largest degree-bounds for the Hamiltonian circuit problem to be easy are 2-in, 2-out. When the degree-bounds increase next to 3-in, 1-out or 3-out, 1-in, the problem becomes NP-complete. References i.

On the line graph of the complete tripartite graph

Abstract: It is shown that the line graph of the complete tripartite graphKn,n,n is characterized by the spectrum of its adjacency matrix.

An extension of Boesch-Thomas optimality in communication nets

Abstract: A \xi -disjoint simultaneous-flow communications net is one whose graph has nonintersecting paths joining (at most) any \xi pairs of its vertices, and is said to be optimal if it also has the least possible number of edges. Such a graph is optimally invulnerable to disconnection in a sense defined previously by Boesch and Thomas, in that its connectivity is equal to the average of the degrees of its vertices. Not all communication nets that are optimal in the Boesch-Thomas sense, however, are optimal in the \xi -disjoint simultaneous-flow sense.

When is the Graph of a Triangulation Uniquely 4-Colorable?

Abstract: (1977). When is the Graph of a Triangulation Uniquely 4-Colorable? The American Mathematical Monthly: Vol. 84, No. 5, pp. 366-367.