Showing papers on "Vertex (graph theory) published in 1984"

Tree graph inequalities and critical behavior in percolation models

Abstract: Various inequalities are derived and used for the study of the critical behavior in independent percolation models. In particular, we consider the critical exponent γ associated with the expected cluster sizex and the structure of then-site connection probabilities τ=τn(x1,..., xn). It is shown that quite generally γ⩾ 1. The upper critical dimension, above which γ attains the Bethe lattice value 1, is characterized both in terms of the geometry of incipient clusters and a diagramatic convergence condition. For homogeneousd-dimensional lattices with τ(x, y)=O(¦x -y¦−(d−2+η), atp=p c, our criterion shows that γ=1 if η> (6-d)/3. The connectivity functions τn are generally bounded by tree diagrams which involve the two-point function. We conjecture that above the critical dimension the asymptotic behavior of τn, in the critical regime, is actually given by such tree diagrams modified by a nonsingular vertex factor. Other results deal with the exponential decay of the cluster-size distribution and the function τ2 (x, y).

On the np‐completeness of certain network testing problems

Abstract: Let G(V, E) be an undirected graph which describes the structure of a communication network. During the maintenance period every line must be tested in each of the two possible directions. A line is tested by assigning one of its endpoints to be a transmitter, the other to be a receiver, and sending a message from the transmitter to the receiver through the line. We define several different models for communication networks, all subject to the two following axioms: a vertex cannot act as a transmitter and as a receiver simultaneously and a vertex cannot receive through two lines simultaneously. In each of the models, two problems arise: What is the maximum number of lines one can test simultaneously? and What is the minimum number of phases necessary for testing the entire network?, where, by “phase” we mean a period in which some tests are conducted simultaneously. We show that in most models, including the “natural” model of radio communication, both problems are NP-hard. In some models the problems can be solved by reducing them to either a maximum matching problem or an edge coloring problem for which polynomial algorithms are known. One model remains for which the complexity of the minimization problem is unknown.

An investigation of the linear three level programming problem

Abstract: The open-loop Stackelberg game is conceptually extended to p players by the multilevel programming problem (MLPP) and can thus be used as a model for a variety of hierarchical systems in which sequential planning is the norm. The rational reaction sets for each of the players is first developed, and then the geometric properties of the linear MLPP are stated. Next, first-order necessary conditions are derived, and the problem is recast as a standard nonlinear program. A cutting plane algorithm using a vertex search procedure at each iteration is proposed to solve the linear three-level case. An example is given to highlight the results, along with some computational experience.

Efficient parallel algorithms for a class of graph theoretic problems

Abstract: In this paper, we present efficient parallel algorithms for the following graph problems: finding the lowest common ancestors for vertex pairs of a directed tree; finding all fundamental cycles, a directed spanning forest, all bridges, all bridge-connected components, all separation vertices, all biconnected components, and testing the biconnectivity of an undirected graph. All these algorithms achieve the $O(\lg ^2 n)$ time bound, with the first two algorithms using $n\lceil n /\lg n\rceil $ processors and the remaining algorithms using $n\lceil n/\lg ^2 n \rceil $ processors. In all cases, our algorithms are better than the previously known algorithms and in most cases reduce the number of processors used by a factor of $n\lg n$. Moreover, our algorithms are optimal with respect to the time-processor product for dense graphs, with the exception of the first two algorithms.The machine model we use is the PRAM which is a SIMD model allowing simultaneous reads but not simultaneous writes to the same memory...

The Multi-Tree Approach To Reliability In Distributed Networks

Abstract: Consider a network of asynchronous processors communicating by sending messages over unreliable lines. There are many advantages to restrict all communications to a spanning tree. To overcome the possible failure of k

On the Maximal Number of Strongly Independent Vertices in a Random Acyclic Directed Graph

Abstract: Let $\mathcal{A}_n $ denote a random acyclic directed graph which is obtained from a random graph with vertex set $\{ 1,2, \cdots ,n\} $, such that each edge is present with a prescribed probability p and all the edges are directed from higher to lower indexed vertices. Define a subset of vertices in $\mathcal{A}_n $ to be strongly independent if there is no directed path between any pair of vertices in the subset. We show that the sequence $\mathcal{J}(\mathcal{A}_n )$, the number of vertices in the largest strongly independent vertex subset of $\mathcal{A}_n $ satisfies with probability tending to 1, \[ \frac{\mathcal{J} (\mathcal{A}_n )}{\sqrt{\log n}} \to \frac{\sqrt{2 }}{\sqrt{\log 1/q}}\quad {\text{as}}\,n \to \infty , \] where $q=1-p$.

Neighborhood complexities and symmetry of chemical graphs and their biological applications

Abstract: Quantitative measures of molecular complexity are calculated through the application of information-theoretic formalism on chemical graphs. The vertex set of a chemical graph is partitioned into disjoint subsets on the basis of the equivalence of various orders of closed neighborhoods and the information indices (ICν, SICν, CIν, and Rν) are calculated. The applications of these indices in structure-activity correlations are discussed.

Complexity of Single-Layer Routing

Abstract: The problem of routing wires between pairs of terminals over a two-dimensional grid is shown to be NP-complete. Actually, the stronger result of routing over any planar graph with no vertex having degree greater than 3 is shown to be NP-complete.

The effect of number of Hamiltonian paths on the complexity of a vertex-coloring problem

Abstract: A generalization of Dobkin and Lipton’s element uniqueness problem is introduced. For any fixed undirected graph G on vertex set $\{ v _1 , v_2 , \cdots , v_n \} $, the problem is to determine, given n real numbers $x_1 ,x_2 , \cdots ,x_n $, whether $x_i e x_j $ for every edge $\{ \upsilon _i ,\upsilon _j \} $ in G. This problem is shown to have upper and lower bounds of $\Theta (n\log n)$ linear comparisons if G is any dense graph. The proof of the lower bound involves showing that any dense graph must contain a subgraph with many Hamiltonian paths, and demonstrating the relevance of these Hamiltonian paths to a geometric argument. In addition, we exhibit relatively sparse graphs for which the same lower bound holds, and relatively dense graphs for which a linear upper bound holds.

Measurable chromatic number of geometric graphs and sets without some distances in euclidean space

Abstract: LetH be a set of positive real numbers. We define the geometric graphG H as follows: the vertex set isR n (or the unit circleS 1) andx, y are joined if their distance belongs toH. We define the measurable chromatic number of geometric graphs as the minimum number of classes in a measurable partition into independent sets. In this paper we investigate the difference between the notions of the ordinary and measurable chromatic numbers. We also prove upper and lower bounds on the Lebesgue upper density of independent sets.

The structures of seven-and twelve-vertex stannacarbaboranes and their relationship to cyclopentadienyl tin cations

Abstract: The stannacarbaboranes, Sn(Me3Si)(Me)C2B4H4(1), Sn(Me)2C2B9H9(2), and (C10H8N2)Sn(Me)2C2B9H9(4) have been prepared and the structures of (1) and (4) have been determined by X-ray crystallography.

Miscellaneous Properties Of Equi-Eccentric Graphs

Abstract: The eccentricity e (v) of a vertex v of a connected graph G is the number maxu∈V(G)d (u, v), where d (u, v) is the distance in G between u and v. A graph is r-equi-eccentric if e (v) = r for every vertex v of G. An r-equi-eccentric graph G of order p is r-minimum if it has the least number of edges of any r-equi-eccentric graph of order p. The following results are given in this paper. (1) A few fundamental properties of equi-eccentric graphs. (2) Several operations for producing equi-eccentric graphs. (3) A characterization of 2-minimum graphs and a proof that the number of edges in a 2-minimum graph of order p is 2p – 5.

Continued fraction methods for random walks on ℕ and on trees

Abstract: where O < r ~ I and a is a nonnegative integer (e.g. [2], [11]). Now the Cayley-graph of a free group F s with s generators is a homogeneous tree of degree s (s edges through each vertex) and so random walks on free groups can be considered as random walks on homogeneous trees. Therefore it seems natural to study random walks on more general trees (or even graphs); this will be done in this paper. In a random walk on a graph a (one-step) transition occurs I from one vertex v to an adjacent vertex with probability d---~' where d(v) is the degree of v (uniform distribution at every vertex). We will be mainly interested in the following quantities: Let 0 be a vertex and

Monochromatic reachability, complementary cycles, and single arc reversals in tournaments

Abstract: Three recent results on tournaments are presented A direct proof is given of the following consequence of a theorem due to B Sands, N Sauer, and R Woodrow [7] : if the arcs of a tournament are two-colored, then there exists some vertex which is reachable from every other vertex via a monochromatic path Next, as illustrative of the proof of a more technical result, it is shown that with one exception every 3-connected tournament contains two complementary cycles And, Adam's Conjecture is established for 2-arc-connected tournaments which are not 3-arc-connected

3-transpositions in infinite groups

Abstract: Let G be a group. A subset D will be called a set of 3-transpositions if | x | = 2 for x e D and | xy | = 3 whenever x, yeD do not commute. We will call the set D closed if xDx −1 = D for each xeD . For each xeD , let For each subset X of D , we denote by [ X ] the graph with vertex set X where two elements x, yeX are joined by an edge whenever they commute. We denote by ( X ) the complement graph; thus two elements x, yeX are joined by an edge of ( X ) whenever they do not commute.

Note on vertex degrees of planar graphs

Abstract: In this note the second moment of the vertex degree sequence of planar graphs is considered. We prove that 1If G is an outerplanar graph of order n ⩾ 3 then 2If G is a planar graph of order n ⩾ 4 then where d1,…,dn is the vertex degree sequence of G. We exhibit all graphs for which these upper bounds are attained.

The conjugacy problem for graph products with central cyclic edge groups

Abstract: A graph product is the fundamental group of a graph of groups. Amongst the simplest examples are HNN extensions and free products with amalgamation. Graph products with cyclic edge groups inherit a solvable conjugacy prob- lem from their vertex groups under certain conditions, the most important of which imposed here is that all the edge group generators in each vertex group are powers of a common central element. Under these conditions the conju- gacy problem is solvable for any two elements not both of zero reduced length in the graph product, and for arbitrary pairs of elements in HNN extensions, tree products and many graph products over finite-leaf roses. The conjugacy problem is not solvable in general for elements of zero reduced length in graph products over graphs with infinitely many circuits. 1. Introduction. A solvable conjugacy problem (S.C.P.) is generally not in- herited by graph products of groups with S.C.P. (see Miller (8)). If attention is re- stricted to graph products with cyclic edge groups, more may be said. It is unlikely that this restriction can be lifted (cf. (6, p. 387; 7, p. 114)). Finite groups, finitely generated free groups, finitely generated nilpotent groups, one-relator groups with torsion or nontrivial centre and certain small cancellation groups all have S.C.P., so there is a wealth of potential vertex groups which may be used in constructing such graph products. In (4) the author shows that a recursively presented graph product with cyclic edge groups over a finite graph inherits S.C.P. from its vertex groups if the sets of cyclic generators in them are "semicritical", thus generalising (5, 7) for HNN extensions and free products with amalgamation. However, semicriticality is a very restrictive condition, which does not hold if all the cyclic generators in a vertex group are powers of a common element. Such cases occur often enough: the celebrated Baumslag-Solitar non-Hopfian groups fall in this category. Here this complementary case is considered. Not surprisingly, a further condition is imposed on the sets of cyclic generators: that they are central in their respective vertex groups. This is suggested by the direct proof that the Baumslag-Solitar groups have S.C.P., and by (2, §3). It is comparatively straightforward to show that the conjugacy problem is solvable for any two elements in the graph product of which at least one has nonzero reduced length (Theorem 3.1). For elements of zero reduced length the problem is much more difficult, reducing to the question of whether a specific recursively enumerable (r.e.) set is recursive (Theorem 3.3).

A Class of Bichromatic Hypergraphs

Abstract: We give a sufficient condition for bichromatic hypergraphs in terms of properties of cycles Application: The set of inclusion-maximal cliques of a perfect graph can be partitioned into two classes such that both classes are represented at every vertex contained in at least two inclusion-maximal cliques

On the reconstruction of rayless infinite forests

Abstract: A graph G is called strongly p-reconstructible if it is (up to isomorphism) uniquely determined by the collection of its pairwise nonisomorphic subgraphs G – v where v is a pendent vertex of G. Using previous results of the second author concerning the structure of infinite rayless graphs it is shown that every rayless forest with an infinite edge-set is strongly p-reconstructible. This result is applied to the classical reconstruction problem to find that every infinite rayless forest G with a finite number of components is reconstructible; i.e., G is (up to isomorphism) uniquely determined by its collection of vertex-deleted subgraphs. Furthermore, an example is given which shows that nonreconstructible rayless forests with a countable number of components exist.

Graphs associated with an integral domain and their applications

Abstract: Publisher Summary This chapter describes the graphs associated with an integral domain and their applications. It is assumed that R is an integral domain of characteristic 0 and Γ a nonempty subset of R \ {0} with the property that α ∈ Γ implies - α ∈ Γ. The simple graph G ( R, Γ ) whose vertex set is R and whose edges are the pairs [ α, β ] satisfying α-β ∉ Γ ( α, s ∈ R ). G ( R,r ) and its complement are Cayley graphs of the additive group of R . It is found that since R is a ring, it is natural to assume that Γ has some multiplicative property. From the point of view of applications, of particular importance is the case when Γ = N · S, where TV is a finite nonempty subset of R and S is a finitely generated multiplicative subsemigroup of R \ {0} with - 1 ∈ S.

Shortest Paths in Signed Graphs

Abstract: A labelling algorithm is proposed to determine the shortest signed paths between one vertex and all others in a weighted signed graph. It can be implemented to take O(m log log D/d) time and O(n + m + D/d) space where D and d denote the lar-the largest and smallest weights of the arcs, respectively. The case where the additional requirement that the paths be elementary is imposed, which is NP-complete, is tackled through decomposition and branch and bound. Applications are made to problems of balance in small groups, transient behaviour of complex societal systems and sign solvability of linear qualitative systems of equations.

Posterior Alpha Activity during Vertex‐CNV Formation under Different Stimulating Conditions

Abstract: In our previous studies, no consistent relationship between vertex contingent negative variation (CNV) elicited by auditory stimuli and parieto-occipital alpha activity (POAA) attenuation (desynchronization) was observed (Zappoli et al., 1977a; Denoth et al., 1979; Zappoli et al., 1980). Hence, we assumed that the CNV complex and the POAA attenuation response are two independent phenomena representing, a t a cortical level, two functional aspects modulated by different ascending diffuse projection multineuronal systems. However, the methods raised doubts about whether the time resolution was adequate to assess relationships between CNV and POAA. POAA was recorded with a constant 8-1 3 Hz digital filter, disregarding the actual pre-S, resting alpha frequency in each trial. This procedure can cause a distortion in envelope detection of alpha activity. The present investigation was conducted using an improved data processing method, one which provides a better analysis of the relationship between the CNV complex, elicited by auditory stimuli (AS) and visual stimuli (VS), and POAA changes.