Frank Harary, Robert Z. Norman, and Dorwin Cartwright. Structural Models: An Introduction to the Theory of Directed Graphs. John Wiley & Sons, 1965.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of other Condensed Graphs. Condensation is a concept used by graph theoreticians for exposing meta level information from a graph by partitioning its vertex set, defining each subset of the partition to be a node in the condensation, and by connecting those nodes according to a well defined rule [6]. Condensed Graphs can thus be represented by a single node (called a condensed node) in a graph at a higher level of abstraction. The basis of the CG firing rule is the presence of a Condensed Graph in every port of a node. That is, a Condensed Graph representing an operand is associated with ....

Harary, F., Norman, R., and Cartwright, D. Structural models: An introduction to the theory of directed graphs. John Wiley and Sons,1969.

....These results provide practical approximation algorithms for NP hard network design problems via an increased understanding of connectivity properties. Until now, the techniques developed have been applicable only to undirected graphs. We consider a basic network design problem in directed graphs [2, 12, 13, 18] which is as follows: given a digraph, find a smallest subset of the edges (forming a minimum equivalent graph (MEG) that maintains all reachability relations of the original graph. When the MEG problem is restricted to strongly connected graphs we call it the minimum SCSS (strongly connected ....

F. Harary, R. Z. Norman and D. Cartwright, Structural models: An introduction to the theory of directed graphs, Wiley, New York, 1965.

....is, in fact, dominated by the second step, i.e. the (n 3 ) time summation and (n 2 ) storage of pair dependencies. This situation is remedied in the next section. The two implementations most widely used to compute betweenness are UCINet [2] and SNAPS. 1 Probably because of a reference to [14] in [11] the latter appear to make use of the following lemma. Recall that the adjacency matrix of a graph is the n n matrix A = a uv ) u;v2V with a uv = 1 if fu; vg 2 E, and a uv = 0 otherwise. Lemma 2 (Algebraic path counting) Let A k = a (k) uv ) u;v2V be the k th power of the ....

F. Harary, R. Z. Norman, and D. Cartwright. Structural Models: An Introduction to the Theory of Directed Graphs. John Wiley & Sons, 1965.

....is converted to the form of a valency matrix V . This matrix is a square matrix of size n, where n is the number of concepts in the corresponding cognitive map. Each element v ij characterizes the relationships between elements i and j. The valency matrix has a number of useful properties [15]. With the valency matrix, we can calculate indirect paths of length 2, 3, 4, etc. Matrices relative to these indirect paths are: V 2 ij = X k v ik v kj 9 V 3 ij = X k v 2 ik v kj etc: For V 2 , each of the terms of the form V ik V kj expresses the indirect effect of a path from ....

F. Harary, R. Z. Norman, and D. Cartwright, Structural Models: An Introduction to the Theory of Directed Graphs, New York: Wiley (1965).

....scheduling problems. The graphs G k and D k should be studied, and properties established. 2. Development of probabilistic analysis methods upon the approach. 3. Examination of sensitivity and or stability for discrete problems on the basis of our approach. Problems of graph stability ([5], 10] are a fundamental for this initiative. 4. Logical functions: Our description of scheduling problems leads to a special class of multi valued logic functions when arguments (n size vector) correspond to permutations and an ordinal scale is used for the objective function. Thus, a ....

F. Harary, R.Z. Norman, and D. Carthwright, Structural Models: An Introduction to the Theory of Directed Graphs. New York: J.Wiley & Sons, 1965.

....faulty node impossible by any algorithm. Hence we assume that the fault propagation graph has been condensed by replacing each strongly connected component of the graph by a single node. Such a condensation of a directed graph with respect to strongly connected components produces an acyclic graph [7]. The condensation can be carried out in linear time in the number of nodes and edges in the original graph. 2.2 Transitive Reduction For the alarm placement problem, we are only interested in knowing whether there is path from one node to another in the graph. A directed graph G t is said to ....

....6. An alarm set A allows unique single fault diagnosis if and only if for every faulty node a 2 V , the set of reachable alarms is nonempty, and for every pair of faulty nodes a; b 2 V , A distinguishes fa; bg. In every acyclic directed graph G, there is at least one node of outdegree 0 [7]. A fault at a node of outdegree 0 does not propagate, and hence an alarm is required in that node in order to achieve fault detection. Besides, every node in the graph G has a directed path to at least one node of outdegree 0 [7] Thus, placing alarms in every node of outdegree 0 ensures that the ....

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F. Harary, R. Z. Norman, and D. Cartwright, Structural Models: An Introduction to the Theory of Directed Graphs (John Wiley, New York, 1965).

.... state aggregation is based upon the Dynamical Consistency (DC) This is a notion developed in the context of forced event (positive imperative) control in [10, 6] generalised to differential control systems in [9] and which, further, are related to notions in a purely graph theoretic setting (see [13]) and in computer science (see [4, 14] An aggregated modular approach to controller reduction also appeared in [27] An analysis of Petri Nets based on refinement and abstraction appeared in [26] with a corresponding analysis of policies that enforce liveness in [25] The present work builds on ....

F. Harary, R.Z. Norman, and D. Cartwright. Structural Models: An introduction to the Theory of Directed Graphs. John Wiley & Sons, New York, 1965.

....abuse of notation, we also use N(x) and N [x] to denote the subgraph induced by these sets. If G 1 = V 1 ; E 1 ) and G 2 = V 2 ; E 2 ) are two graphs, we will denote G 1 [ G 2 = V 1 [ V 2 ; E 1 [ E 2 ) The union of two digraphs is defined accordingly. Theorem 1 (Harary, Norman and Cartwright [3]) Let D = V; A) be a digraph. Then D is acyclic if and only if there exists an ordering of vertices, oe = v 1 ; v 2 ; Delta Delta Delta ; v n ] such that one of the following two conditions holds, 1) For all i; j 2 [1; n] v i ; v j ) 2 A implies that i j. 2) For all i; j 2 [1; n] v ....

F. Harary, R.Z. Norman and D. Cartwright, Structural Models: An Introduction to the Theory of Directed graphs, John Wiley, New York, 1965.

....of a partition machine permits a natural construction of a large class of hierarchical control structures on any given finite state machine. It is to be noted that in the purely graph theoretic setting, without any controlled dynamics, an idea related to that of DC dynamics is to be found in [5]. We note that there is a natural parallel between the formulation of levels in hierarchical system theory and their definition in hybrid system theory, where discrete systems play the roles of both models and controllers for finer continuous state systems. The reader is referred to [2] and [4] ....

F. Harary, R.Z. Norman, and D. Cartwright. Structural Models: An introduction to the Theory of Directed Graphs. John Wiley & Sons, New York, 1965.

....systems. Many shortest path oriented models, such as those dedicated to best routing in transport networks, require directed graphs, that is, graphs in which edges have directions. We adopt for the most part the terminology and notation of Deo [29] which are commonly found in the literature (see [25, 26, 30, 63, 64], for instance) A directed graph or digraph consists of a finite set V = fv k g n k=1 of vertices and a set A = fa l g m l=1 of ordered pairs of vertices called arcs, where a l = v s(l) v t(l) with s(l) being the index of the source or initial vertex of the l th arc and t(l) the index of ....

F. Harary, R.Z. Norman and D. Cartwright, Structural Models: An Introduction to the Theory of Directed Graphs, John Wiley & Sons, Inc., New York, 1965.

....These results provide practical approximation algorithms for NP hard network design problems via an increased understanding of connectivity properties. Until now, the techniques developed have been applicable only to undirected graphs. We consider a basic network design problem in directed graphs [2, 12, 13, 18] which is as follows: given a digraph, find a smallest subset of the edges (forming a minimum equivalent graph (MEG) that maintains all reachability relations of the original graph. When the MEG problem is restricted to strongly connected graphs we call it the minimum SCSS (strongly connected ....

F. Harary, R. Z. Norman and D. Cartwright, Structural models: An introduction to the theory of directed graphs, Wiley, New York, 1965.

....represented as a partial order on variables, labelled by first order quantifiers. Interpretable formulas of L(Ivar) can also be represented as a labelled structure on variables, but in this case the structure is not always a partial order. First we introduce some graph theoretic terminology (see Harary, Norman and Cartwright [1965]) A digraph (from directed graph) is a pair D = V;A , where V is the set of vertices and A is a set of ordered pairs u; v , where u 6= v 2 V ; elements of A are called arcs. u is a transmitter in D if there is no v such that v; u 2 A; u is a receiver if there is no v such that u; v 2 A. A ....

.... if v is a vertex such that all u with u; v 2 A have been labelled, label v by x indexed by the labels of immediate predecessors of v (where x is new) If t is the label of the receiver, D is isomorphic to D(t) the assumption that D is acyclic guarantees that each vertex will be labelled (cf. Harary et al. [1965], theorem 10.1) More generally, we may view prefixes as acyclic digraphs labelled by quantifiers. Take a prefix P and construct a digraph D(P ) V;A by V is the set of variables which hereditarily occur in P A = f u; v ju; v 2 V; v immediately depends on ug. If we label D(P ) in the obvious ....

F. Harary, R. Z. Norman and D. Cartwright [1965]. Structural models: an introduction to the theory of directed graphs, Wiley.

....of a partition machine permits a natural construction of a large class of hierarchical control structures on any given finite state machine. It is to be noted that in the purely graph theoretic setting, without any controlled dynamics, an idea related to that of DC dynamics is to be found in [5]. We note that there is a natural parallel between the formulation of levels in hierarchical system theory and their definition in hybrid system theory, where discrete systems play the roles of both models and controllers for finer continuous state systems. The reader is referred to [2] and [4] ....

F. Harary, R.Z. Norman, and D. Cartwright. Structural Models: An introduction to the Theory of Directed Graphs. John Wiley & Sons, New York, 1965.

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Frank Harary, Robert Z. Norman, and Dorwin Cartwright. Structural Models: An Introduction to the Theory of Directed Graphs. John Wiley & Sons, 1965.

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F. Harary, Structural Models: An Introduction to the Theory of Directed Graphs. New York: Wiley, 1965.

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Frank Harary, Robert Z. Norman, and Dorwin Cartwright. Structural Models: An Introduction to the Theory of Directed Graphs. John Wiley & Sons, 1965.

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F. Harary, R. Norman, and D. Cartwight. Structural Models : an introduction to the theory of directed graphs. Wiley, 1965.

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F. Harary, R. Z. Norman, and D. Cartwright, Structural Models: An Introduction to the Theory of Directed Graphs. New York: Wiley, 1965.

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HARARY, F, NORMAN, R.Z., and CARTWRIGHT, D. (1965), Structural Models: An Introduction to the Theory of Directed Graphs, New York: John Wiley & Sons.

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Harary, F., R.Z. Norman and D. Cartwright (1965), Structural Models : An Introduction to the Theory of Directed Graphs, Macmillan, New York, London, Sidney.

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F. Harary, and R.Z. Norman, and D. Carthwright, Structural Models: An Introduction to the Theory of Directed Graphs, New York: J. Wiley & Sons, 1995. 16

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F. Harary, R. Z. Norman, and D. Cartwright. Structural Models: An Introduction to the Theory of Directed Graphs. John Wiley & Sons, New York, 1965.

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