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Showing papers on "Disjoint sets published in 1983"


Book
Robert E. Tarjan1
01 Jan 1983
TL;DR: This paper presents a meta-trees tree model that automates the very labor-intensive and therefore time-heavy and therefore expensive process of manually selecting trees to grow in a graph.
Abstract: Foundations Disjoint Sets Heaps Search Trees Linking and Cutting Trees Minimum Spanning Trees Shortest Paths Network Flows Matchings

2,120 citations


Journal ArticleDOI
TL;DR: A sorting network withcn logn comparisons where in thei-th step of the algorithm the contents of registersRj, andRk, wherej, k are absolute constants then change their contents or not according to the result of the comparison.
Abstract: We give a sorting network withcn logn comparisons. The algorithm can be performed inc logn parallel steps as well, where in a parallel step we comparen/2 disjoint pairs. In thei-th step of the algorithm we compare the contents of registersR j(i) , andR k(i) , wherej(i), k(i) are absolute constants then change their contents or not according to the result of the comparison.

497 citations


Proceedings ArticleDOI
01 Dec 1983
TL;DR: A linear-time algorithm for the special case of the disjoint set union problem in which the structure of the unions (defined by a “union tree”) is known in advance, which gives similar improvements in the efficiency of algorithms for solving a number of other problems.
Abstract: This paper presents a linear-time algorithm for the special case of the disjoint set union problem in which the structure of the unions (defined by a “union tree”) is known in advance. The algorithm executes an intermixed sequence of m union and find operations on n elements in 0(m+n) time and 0(n) space. This is a slight but theoretically significant improvement over the fastest known algorithm for the general problem, which runs in 0(ma(m+n, n)+n) time and 0(n) space, where a is a functional inverse of Ackermann's function. Used as a subroutine, the algorithm gives similar improvements in the efficiency of algorithms for solving a number of other problems, including two-processor scheduling, the off-line min problem, matching on convex graphs, finding nearest common ancestors off-line, testing a flow graph for reducibility, and finding two disjoint directed spanning trees. The algorithm obtains its efficiency by combining a fast algorithm for the general problem with table look-up on small sets, and requires a random access machine for its implementation. The algorithm extends to the case in which single-node additions to the union tree are allowed. The extended algorithm is useful in finding maximum cardinality matchings on nonbipartite graphs.

398 citations


Journal ArticleDOI
TL;DR: It is shown that any embedding of K7 in three-dimensional euclidean space contains a knotted cycle and that any embedded cycle of K6 contains a pair of disjoint cycles which are homologically linked.
Abstract: The main purpose of this paper is to show that any embedding of K7 in three-dimensional euclidean space contains a knotted cycle. By a similar but simpler argument, it is also shown that any embedding of K6 contains a pair of disjoint cycles which are homologically linked.

298 citations


Journal ArticleDOI
TL;DR: In this article, the Torelli group was shown to be finitely generated for both gg 0 and fgg 0 when gg ≥ 3 and a simple set of generators was given.
Abstract: This is the first of three papers concerning the so-called Torelli group. Let M = Mg be a compact orientable surface of genus g having n boundary components and let 9 = Xg . be its mapping class group, that is, the group of orientation preserving diffeomorphisms of M which are 1 on the boundary AM modulo isotopies which fix 3M pointwise. This group is also known to the complex analysts as the Teichmuller group or modular group. If n = 0 or 1, let further 4 = Jg . be the subgroup of D1 which acts trivially on H1(M, Z). The topologists have no traditional name for A, but the analysts tell me it was known classically and is called the Torelli group. Several interesting problems and conjectures exist concerning f. The principal one can be found in Kirby's problem list [K] and asks if gg is finitely generated. In this first paper we shall answer the question affirmatively for both gg 0 and fgg , when g > 3 and shall give a fairly simple set of generators. Two other conjectures were made by the author. The first involves the subgroup 'J of f which is generated by twists on nulhomologous simple closed curves. [JI] produces a surjective homomorphism T: fgg 1 A3H1(M, Z) which kills C, and it is conjectured there that in fact 5Y = Ker T. The proof of this is the content of the second paper. In the third paper we use the results of the second to compute the abelianization f/f' explicitly, thereby verifying another conjecture in [Ji]. The first reasonably simple (but infinite) set of generators for fg0 was produced by Powell in [P]. His generators consist of two types: a) twists on bounding simple closed curves, b) opposite twists on a (bounding) pair of disjoint homologous simple closed curves, each of which are nonbounding. Using his result, the author showed in [J2] that the maps of the second type, which we call BP maps (for bounding pair), are actually sufficient to generate both Kg 0 and fg 1 for g > 3 and in fact that we need only those whose two curves bound a genus one subsurface of M (note that for g = 2 all BP maps are trivial and hence the result fails in this case). In the finite set of generators produced in this paper only

283 citations


Book ChapterDOI
01 Jan 1983
TL;DR: This work describes an algorithm due to Kronecker based on the minimum root Separation, Sturm's algorithm, an algorithm based on Rolle’s theorem due to Collins and Loos and the modified Uspensky algorithm dueto Collins and Aritas.
Abstract: Let A be a polynomial over Z, Q or Q(α) where α is a real algebraic number. The problem is to compute a sequence of disjoint intervals with rational endpoints, each containing exactly one real zero of A and together containing all real zeros of A. We describe an algorithm due to Kronecker based on the minimum root Separation, Sturm’s algorithm, an algorithm based on Rolle’s theorem due to Collins and Loos and the modified Uspensky algorithm due to Collins and Aritas. For the last algorithm a recursive version with correctness proof is given which appears in print for the first time.

148 citations


Journal ArticleDOI
Tom Hudson1
TL;DR: Hudson et al. as discussed by the authors found that young children are skillful at establishing correspondences and determining exact numerical differences between disjoint sets; their poor performance on the standard questions apparently reflects a misinterpretation or inadequate comprehension of comparative constructions of the general form "How many... [comparative term].. than...?"
Abstract: HUDSON, TOM. Correspondences and Numerical Differences between Disjoint Sets. CHILD DEVELOPMENT, 1983, 54, 84-90. Young children's understanding of correspondences and numerical differences between disjoint sets was studied in a series of 3 experiments. In the first 2 experiments, 64 children between 4 and 8 years of age were shown pairs of sets and were asked both standard ("How many more birds than worms are there?") and nonstandard ("How many birds won't get a worm?") numerical difference questions. The children's observed success in answering the Won't Get questions indicates that many young children are skillful at establishing correspondences and determining exact numerical differences between disjoint sets; their poor performance on the standard questions apparently reflects a misinterpretation or inadequate comprehension of comparative constructions of the general form "How many . . . [comparative term] . . than . .. ?" The final experiment, involving 30 additional kindergarten children, dealt with children's solution strategies in answering Won't Get questions. The most frequently observed solution strategy was a sophisticated indirect counting strategy rather than a perceptually guided pairing strategy. Taken together, the present findings restrict the domain of applicability of the theory that young children are limited to perceptually based forms of mathematical reasoning.

143 citations


Journal ArticleDOI
TL;DR: In this paper, a disjoint Latin square of order n with n parallel transversals including the diagonal one is constructed, which is called LDS(n) and is a set of n + 2 pairwise disjunctional Latin squares.

140 citations


Journal ArticleDOI
Dan Gusfield1

129 citations


Journal ArticleDOI
TL;DR: A sequence of m consecutive positive integers is said to be perfect if the integers can be arranged in disjoint pairs {(a"i, b"i): 1= 1=}.

110 citations


Journal ArticleDOI
TL;DR: The condition λv(v − 1) ≡ 0 (mod 2m), v ⩾ m + 1 is obviously necessary for the existence of an edge disjoint decomposition of a complete multigraph λKv into isomorphic simple paths consisting of m edges each.

Journal ArticleDOI
01 Mar 1983
TL;DR: For a finite ordered set P, the greedy algorithm was shown to be optimal for a far wider class of ordered sets than was hitherto suspected in this paper, where the authors showed that a natural greedy algorithm is actually optimal for ordered sets.
Abstract: For a finite ordered set P how can a linear extension L = Cl D C2 D... Cm,, be constructed which minimizes the number m of chains C, of P? While this question remains largely unanswered we show that a natural "greedy" algorithm is actually optimal for a far wider class of ordered sets than was hitherto suspected. The most fundamental and yet far-reaching results in the theory of ordered sets are the following: (R. P. Dilworth [1950]) In a finite ordered set P, the minimum number of disjoint chains whose set union is all of P equals the maximum number of pairwise noncomparable elemnents of P. (E. Szpilrajn [1930]) Every ordered set has a linear extension. Let C1, C2,. . . ,CCm be any minimum family of disjoint chains of a finite ordered set P whose set union is P. The linear sum C1 e C2 e ... e Cm of these chains need not be a linear extension of P. On the other hand, any linear extension L of a finite ordered set P can be expressed as the linear sum C1 e C2 3 * * Cm of chains Ci in P (so chosen that supp Cl 4 infp Cl+ 1 for each i) and, while Uk I Ci = P, this family of chains need not be the minimum number whose set union is P (see Figure 1). How can these two fundamental results about ordered sets be reconciled? Indeed, until recently there seemed little reason even to consider such a question.

Journal ArticleDOI
TL;DR: It is shown that, for each natural numberk, these exists a (smallest) natural numberf(k) such that any digraph of minimum outdegree at leastf( k) containsk disjoint cycles.
Abstract: We show that, for each natural numberk, these exists a (smallest) natural numberf(k) such that any digraph of minimum outdegree at leastf(k) containsk disjoint cycles. We conjecture thatf(k)=2k−1 and verify this fork=2 and we show that, for eachk≧3, the determination off(k) is a finite problem.

Journal ArticleDOI
TL;DR: In this paper, the evolution of a two-dimensional, incompressible, ideal fluid in a case in which the vorticity is concentrated in small, disjoint regions of the physical space is studied.
Abstract: We study the evolution of a two dimensional, incompressible, ideal fluid in a case in which the vorticity is concentrated in small, disjoint regions of the physical space. We prove, for short times, a connection between this evolution and the vortex model.

Journal ArticleDOI
TL;DR: In this paper, it was shown that the mean square fluctuation in the net electric charge contained in a subregion of an infinitely extended equilibrium Coulomb system (plasma, electrolytes, etc) grows only as the surface area of the system grows.
Abstract: It was shown by Martin and Yalcin that the mean-square fluctuation $〈{Q}_{\ensuremath{\Lambda}}^{2}〉$ in the net electric charge ${Q}_{\ensuremath{\Lambda}}$ contained in a subregion $\ensuremath{\Lambda}$ of an infinitely extended equilibrium Coulomb system (plasma, electrolytes, etc) grows only as the surface area ${S}_{\ensuremath{\Lambda}}$ (not the volume) of $\ensuremath{\Lambda}$ and that $\frac{{Q}_{\ensuremath{\Lambda}}}{\sqrt{{S}_{\ensuremath{\Lambda}}}}$ has a Gaussian distribution as $\ensuremath{\Lambda}\ensuremath{\rightarrow}\ensuremath{\infty}$ We extend these results to joint charge fluctuations in different spatial regions: Let space be divided into disjoint regions ${\ensuremath{\Lambda}}_{i}$, $i=1,2,\dots{}$, say, cubes of length $L$ We show that as $L\ensuremath{\rightarrow}\ensuremath{\infty}$, the covariance in $\frac{{Q}_{{\ensuremath{\Lambda}}_{i}}}{L}$ behaves as ${L}^{\ensuremath{-}2}〈{Q}_{{\ensuremath{\Lambda}}_{i}}{Q}_{{\ensuremath{\Lambda}}_{j}})=\ensuremath{-}\frac{1}{6}{L}^{\ensuremath{-}2}〈{Q}_{{\ensuremath{\Lambda}}_{i}}^{2}〉=\ensuremath{-}\frac{1}{6}K$ if ${\ensuremath{\Lambda}}_{i}$ and ${\ensuremath{\Lambda}}_{j}$ are adjacent, and is zero if they do not have a common face Furthermore, the variables $\frac{{Q}_{\ensuremath{\Lambda}}}{L}$ approach, as $L\ensuremath{\rightarrow}\ensuremath{\infty}$, a jointly Gaussian distribution These results can be proven rigorously whenever the correlations in the system decay faster than the fourth power of the distance, which is known to happen in many cases This behavior of charge fluctuations is shown to be required for the consistency of the usual statistical-mechanical treatment of neutralmolecular systems

Journal ArticleDOI
01 Sep 1983-Networks
TL;DR: A method to determine a maximum set of crossings (intersections of segments) with no two on the same segment, as well as amaximum set of nonintersecting segments, both in O(n3/2 log2 log 2 log2n) time are given.
Abstract: Let S be a set of n horizontal and vertical segments on the plane, and let s, t ∈ S. A Manhattan path (of length k) from s to t is an alternating sequence of horizontal and vertical segments s = r0, r1,…,rk = t where ri intersects ri+1, 0 ≤ i < k. An from all s ∈ S to t. Also given is a method to determine a maximum set of crossings (intersections of segments) with no two on the same segment, as well as a maximum set of nonintersecting segments, both in O(n3/2 log2 log2n) time. The latter algorithm is applied to decomposing, in O(n3/2 log2n) time, a hole-free union of n rectangles with sides parallel to the coordinate axes into the minimal number of disjoint rectangles. All the algorithms require O(n log n) space, and for all of them the factor log2n can be improved to log n log log n, at the cost of some complication of the basic data structure used.

Journal ArticleDOI
TL;DR: Jorgensen et al. as mentioned in this paper showed that a set of elements representing simple disjoint loops on U/G can be made parabolic on the boundary of G, where the deformations are all supported on U. The restriction to function groups is to some extent a matter of convenience; their techniques can be applied in certain more general situations.
Abstract: Let G be a Kleinian group, and T(G) its deformation space. We are concerned here with the question of which loxodromic (including hyperbolic) elements of G can be made parabolic on the boundary of T(G). For some geometrically finite boundary groups (see [8]), one obtains necessary conditions in terms of primitive elements of G represented by simple disjoint loops on U(G)/G. In this paper we restrict our attention to geometrically finite function groups (including groups with torsion), and show that these necessary conditions are sufficient. The restriction to function groups is to some extent a matter of convenience; our techniques can be applied in certain more general situations. The groups we obtain as boundary groups are all geometrically finite. Nothing is known about which elements can be made parabolic at boundary groups which are not geometrically finite. If G is Fuchsian acting on the upper half plane U, then every element of G representing a simple loop on U/G is primitive. Abikoff [1] and Marden, in the torsion-free case (unpublished), showed that a set of elements representing simple disjoint loops on U/G can be made parabolic on the boundary of the deformation space of G, where the deformations are all supported on U. Similar results along the lines of this paper have also been obtained by Thurston (unpublished). The author wishes to thank T. Jorgensen, L. Keen and I. Kra for informative conversations.

Journal ArticleDOI
TL;DR: In this paper, an algorithm for the construction of equine-ighbored balanced incomplete block designs with k = 3$ is presented, which makes use of the decomposition of complete graphs into disjoint Hamiltonian cycles.
Abstract: Some methods for the construction of equineighbored balanced incomplete block designs introduced by Kiefer and Wynn (1981) are presented. An algorithm for constructing designs with $k = 3$ is developed. Kiefer and Wynn's result for $k = 3$ is difficult to implement in practice. Our algorithm provides a practical solution and makes use of the decomposition of complete graphs into disjoint Hamiltonian cycles. The construction of designs with $k = v - 1$ and $v - 2$ is also completely solved. The neighbor designs proposed for use in serology are useful for the construction of equineighbored balanced incomplete block designs. Several infinite families of equineighbored balanced incomplete block designs are listed.

Journal ArticleDOI
TL;DR: This work considers sets of MOLS (mutually orthogonal Latin squares) having holes, corresponding to missing sub-MOLS, which are disjoint and spanning, and gives several constructions for sets with holes.

Journal ArticleDOI
TL;DR: This is the first known example of a complete partition of P k ( v ) into disjoint S ( t, k , v )'s for k ⩽4 and t ⩾2.

Journal ArticleDOI
TL;DR: In this paper, the authors construct two non-compact σ compact spaces X, one locally compact and one nowhere locally compact, such that X has no remote points, and in fact such that βX is not extremally disconnected at any point.
Abstract: All spaces considered are completely regular and X* denotes βX — X. The point x G X* is called a remote point of X if x g C\βxA for each nowhere dense subset A of X. If y G 7, then the space Y is said to be extremally disconnected at y if j> £ ί/ Π F whenever £/and Fare disjoint open sets. In this paper we construct two noncompact σ-compact spaces X, one locally compact and one nowhere locally compact, such that X has no remote points, and in fact such that βX is not extremally disconnected at any point.

Journal ArticleDOI
Geoffrey Exoo1
TL;DR: In a recent paper Lovasz, Neumann-Lara and Plummer proved some Mengerian theorems for paths of bounded length, the line connectivity analogue of their problem is considered.

Journal ArticleDOI
01 Feb 1983
TL;DR: In this paper, it was shown that the completely regular, monotone maps arising as quotient maps of these decompositions are cell-like maps, and that such maps can be extended to higher dimensions.
Abstract: Certain decompositions of homogeneous continua are shown to be cell-like. In particular, the aposyndetic decomposition described by F. B. Jones of a homogeneous, decomposable continuum is cell-like, and we prove that any homogeneous decomposable continuum admits a continuous decomposition into mutually homeomorphic, indecomposable, homogeneous, cell-like terminal continua so that the quotient space is an aposyndetic homogeneous continuum. D. C. Wilson [8] has shown that a monotone, completely regular map f: X -p Z of the n-dimensional continuum X onto the nondegenerate continuum Y has the property that Hn(f-1(z)) = 0, for all z in Z. In particular, if n 1, then the point inverses of f are acyclic continua. More recently, Mason and Wilson [5] have shown that if n = 1, then the point inverses of f are tree-like continua, that is, f is a cell-like map. Since the projection maps of a product space onto the factors are completely regular maps, one cannot, in general, extend the Mason-Wilson result to higher dimensions. The impetus for the Mason-Wilson result, however, was the author's applications [6, 7] of completely regular maps to certain monotone decompositions of homogeneous continua. In this paper, we show that the completely regular, monotone maps arising as quotient maps of these decompositions are cell-like maps. In particular, Jones' aposyndetic decomposition of a homogeneous, decomposable continuum is a decomposition of that continuum into cell-like sets. A continuum X is cell-like if each mapping of X into a compact ANR is inessential. If the map f is inessential, we write f 0 O. A continuum is cell-like if and only if it has trivial shape. A map is cell-like if each of its point inverses is cell-like. The following theorem is classical. WIRECUTTING THEOREM. Let A and B be closed subsets of the compact space M. If no connected subset of M intersects both A and B, then there exist disjoint closed subsets M1 and M2 of M such that A c M1, B c M2, and M = M1 U M2. A subcontinuum Z of the continuum X is said to be terminal if each subcontinuum Y of X such that Y n Z # 0 satisfies either Y c Z or Z c Y. The proof of the next theorem is similar to that of [2, Theorem 2]. Received by the editors September 11, 1981. 1980 Mathematics Subject Classification. Primary 54F20, 54F50.

Journal ArticleDOI
01 Apr 1983
TL;DR: Agarwal et al. as mentioned in this paper showed that the countable chain condition and metrizability are equivalent for Gul'ko compact sets, and proved that a compact space Ai' is said to be Gul'kos compact if the space C( K ) is ^X-countably determined in the weak topology.
Abstract: A compact space Ai' is said to be Gul'ko compact if the space C( K ) is ^X-countably determined in the weak topology. Well-known compact sets, such as Eberlein compact sets, are Gul'ko compact. We prove here that the countable chain condition and metrizability are equivalent for Gul'ko compact sets. The purpose of this note is to give a positive answer to a question of Talagrand (Probleme 7.9 in [13]). In fact, we prove that: if A is a compact Hausdorff space, such that the Banach space C(A) of all real-valued continuous functions on A is ^countably determined in its weak topology, then in A there is a family of pairwise disjoint nonempty open sets of cardinality equal to the least cardinality of a base for A. We denote by 2 the space of irrationals; 2 is identified with NN. 5 denotes the set of finite sequences of natural numbers. For s E S we denote by | s | the length (i.e.. the domain) of s. For s, t E S, we write i ,er G [0,l]r. In a Corson-compact space A, the weight of A (i.e., the least cardinality of a base for A) is equal to the density character of A (i.e., to the least cardinality of a dense subset of A ). Furthermore, every sequence in A has a convergent subsequence. A topological space A satisfies the countable chain condition (c.c.c.) if every family of pairwise disjoint nonempty open subsets of A is countable (cf. [5]). The following definition (in some equivalent form) has been introduced by Archangel'skii [2], VaSak [14], Definition. A topological space A is 5(-countably determined if there are a subset 2' of the space 2 of irrationals, a compact Hausdorff space A, and a closed subset £ of A X 2' such that (denoting by 77,: A X 2' -> A the projection w,(x, y) = x) A=7T,(£). _ Received by the editors May 7, 1982 and, in revised form, July 26. 1982. 1980 Mathematics Subject Classification. Primary 54E65, 54H05; Secondary 46B99. ©1983 American Mathematical Society OOO2-9939/82/O000-0774/$02.OO 731 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 732 s. ARGYROS AND S NEGREPONTIS If 2' = 2, then A is called %-analytic (cf. [4]). We need the following simple Lemma. Let X be a "h-countably determined space. There is a family {A^: s E S) of subsets of X such that Aa — X, Uk<.uAfii — As for s E S, and for every x E X there is a a El. such that (a) x G H^^/l^ and (b) // xk E An]k for k < to then the sequence ( xk ) has a limit point in X. Proof. There are 2' C 2, a compact space A, and a closed subset £ of A x 2' such that 77,(£) = X. There is a family {Bs: s E S) of subsets of 2' such that B0 — 2', UA

Journal ArticleDOI
TL;DR: In this article, a rank-one quasi-Newton update class is introduced, which is a subset of the generalized Huang-Oren class and contains two disjoint subclasses of positive definite updates.
Abstract: We introduce variationally a class of rank-one Quasi-Newton updates, which is a subset of the generalized Huang-Oren class. We show that this class contains two disjoint subclasses of positive definite updates. One optimally conditioned update in the sense of Oren and Spedicato is shown to exist in each of the two subclasses. Some criteria for selection of the remaining degrees of freedom and some numerical experiments are discussed.

01 Jan 1983
TL;DR: In this article, it was shown that the unique near hexagon with s = 2 and t = 14 and t 2 = 2 is the one with the blocks of the Steiner system S(5,8,24) as vertices and sets of three pairwise disjoint blocks as lines.
Abstract: We show that the unique near hexagon with s = 2 and t = 14 and t 2 = 2 is the one with the blocks of the Steiner system S(5,8,24) as vertices and sets of three pairwise disjoint blocks as lines.

Journal ArticleDOI
01 Sep 1983-Networks
TL;DR: Methods for finding set colorings, phasings, and intersection assignments in which the measure of the union of the intervals S(x) is minimized or the sum of the lengths of the S( x) is maximized are presented.
Abstract: This paper studies set assignments on graphs, functions assigning a set S(x) to each vertex x of a graph, and specifically set assignments where each set is a real interval, perhaps of specified minimum length. Such set assignments arise in applied problems dealing with fleet maintenance, mobile radio frequency assignment, task assignment, traffic phasing, banquet preparation, and computer storage optimization. These problems are briefly discussed. They are translated into problems of finding a set coloring [a set assignment in which an edge between x and y implies that S(x) and S(y) are disjoint], a set phasing [a set coloring of the complementary graph], or a set intersection assignment. The paper presents methods for finding set colorings, phasings, and intersection assignments in which the measure of the union of the intervals S(x) is minimized or in which the sum of the lengths of the S(x) is maximized.

Journal ArticleDOI
01 Mar 1983
TL;DR: In this article, it was shown that the product of countably many scattered paracompact spaces is even ultra-paraccompact, i.e., it has a disjoint, topped, open refinement (covering Y).
Abstract: In this paper we prove that the product of countably many scattered paracompact spaces is even ultraparacompact. Telgarsky [1] has shown that scattered paracompact spaces are ultraparacompact. Verbally, H. Martin has asked if a product of countably many spaces with exactly one nonisolated point has to be paracompact. We prove THEOREM. The product of countably many scattered paracompact spaces is ultraparacompact. All spaces are assumed Hausdorff. A space is ultraparacompact if every open cover has a disjoint open refinement. We occasionally use the word refinement when less than the whole space is covered: if so the covered subspace is always mentioned. A scattered space Xis U,< Xa for some minimal ordinal A where, for a < X, Xa is the set of all isolated points of XU

Journal ArticleDOI
TL;DR: In the disjoint products version of the cut-based method of reliability analysis of graphs it is necessary to determine the order in which the cuts are to be considered as mentioned in this paper.
Abstract: In the disjoint products version of the cut-based method of reliability analysis of graphs it is necessary to determine the order in which the cuts are to be considered. The usual approach has been to order them in terms of the number of arcs in the cut, with the cuts with the least number of arcs considered first. However, the reliability formula can be simplified if the cuts are considered in an order which is obtained by considering the node partition associated with each cut. There does not seem to be any such ordering of the paths that will enable a corresponding simplification of the minpath formula.

Journal ArticleDOI
Cai Mao-cheng1
TL;DR: In this article, the authors give a necessary and sufficient condition for a digraph G to contain k arc-disjoint arborescences so that the number rooted at each vertex x of G lies in some prescribed interval which depends on x.
Abstract: The purpose of this paper is to give a necessary and sufficient condition for a digraph G to contain k arc-disjoint arborescences so that the number rooted at each vertex x of G lies in some prescribed interval which depends on x