On partitions of functions on finite sets

Let D and R be finite sets with cardinality n and m respectivelyR ^D be the set of all functions from D into R, and G and H be permutation groups acting on D and R respectively. Two functions f and g in R ^D are said to be related if there exists a σ in G and a τ in H with f(σd) = τg(d) for every d in D. Since the relation is an equivalence relation, R ^D is partitioned into disjoint classes. Roughly, by using the cycle indices of G and H, de Bruijn's theorem determines the number of equivalence classes, and Pólya's theorem, with H being the identity group, gives the function counting series, Pólya-de Bruijn's theorem has many applications (for instance, see Pólya and Read [6]). The theorem and its applications, basically, centered around the partitions of functions. Here, we present an algorithm to determine which functions in R ^D belong to the same equivalent class. Our algorithm does not use the cycle indices of G and H (to compute the cycle index of a given group, in general, is difficult), but it uses the generators of G and H, and the m-nary numbers to code the functions in R ^D . Our algorithm also gives the function counting series and the number of equivalence classes. An important application is that for each positive integer n, we use our algorithm and the symmetric group S _n to determine all isomorphic and nonisomorphic graphs and directed graphs with n vertices.     

Rank and eigenvalues of a supersymmetric tensor, the multivariate homogeneous polynomial and the algebraic hypersurface it defines

A real n-dimensional homogeneous polynomial f(x) of degree m and a real constant c define an algebraic hypersurface S whose points satisfy f(x)=c. The polynomial f can be represented by Ax^m where A is a real mth order n-dimensional supersymmetric tensor. In this paper, we define rank, base index and eigenvalues for the polynomial f, the hypersurface S and the tensor A. The rank is a nonnegative integer r less than or equal to n. When r is less than n, A is singular, f can be converted into a homogeneous polynomial with r variables by an orthogonal transformation, and S is a cylinder hypersurface whose base is r-dimensional. The eigenvalues of f, A and S always exist. The eigenvectors associated with the zero eigenvalue are either recession vectors or degeneracy vectors of positive degree, or their sums. When c⁄=0, the eigenvalues with the same sign as c and their eigenvectors correspond to the characterization points of S, while a degeneracy vector generates an asymptotic ray for the base of S or its conjugate hypersurface. The base index is a nonnegative integer d less than m. If d=k, then there are nonzero degeneracy vectors of degree k−1, but no nonzero degeneracy vectors of degree k. A linear combination of a degeneracy vector of degree k and a degeneracy vector of degree j is a degeneracy vector of degree k+j−m if k+j≥m. Based upon these properties, we classify such algebraic hypersurfaces in the nonsingular case into ten classes.     

Toughness of the corona of two graphs

The toughness of a non-complete graph G=(V, E) is defined as τ(G)=min{|S|/ω(G?S)}, where the minimum is taken over all cutsets S of vertices of G and ω(G?S) denotes the number of components of the resultant graph G?S by deletion of S. The corona of two graphs G and H, written as G° H, is the graph obtained by taking one copy of G and |V(G)| copies of H, and then joining the ith vertex of G to every vertex in the ith copy of H. In this paper, we investigate the toughness of this kind of graphs and obtain the exact value for the corona of two graphs belonging to some families as paths, cycles, stars, wheels or complete graphs.     

Vanquishing the XCB Question: The Methodological Discovery of the Last Shortest Single Axiom for the Equivalential Calculus

With the inclusion of an effective methodology, this article answers in detail a question that, for a quarter of a century, remained open despite intense study by various researchers. Is the formula XCB=e(x,e(e(e(x,y),e(z,y)),z)) a single axiom for the classical equivalential calculus when the rules of inference consist of detachment (modus ponens) and substitution Where the function e represents equivalence, this calculus can be axiomatized quite naturally with the formulas e(x,x), e(e(x,y),e(y,x)), and e(e(x,y),e(e(y,z),e(x,z))), which correspond to reflexivity, symmetry, and transitivity, respectively. (We note that e(x,x) is dependent on the other two axioms.) Heretofore, thirteen shortest single axioms for classical equivalence of length eleven had been discovered, and XCB was the only remaining formula of that length whose status was undetermined. To show that XCB is indeed such a single axiom, we focus on the rule of condensed detachment, a rule that captures detachment together with an appropriately general, but restricted, form of substitution. The proof we present in this paper consists of twenty-five applications of condensed detachment, completing with the deduction of transitivity followed by a deduction of symmetry. We also discuss some factors that may explain in part why XCB resisted relinquishing its treasure for so long. Our approach relied on diverse strategies applied by the automated reasoning program OTTER. Thus ends the search for shortest single axioms for the equivalential calculus.     

A Decision Procedure for Linear “Big O” Equations

Let F be the set of functions from an infinite set, S, to an ordered ring, R. For f, g, and h in F, the assertion f = g + O(h) means that for some constant C, |f(x) − g(x)| ≤C |h(x)| for every x in S. Let L be the first-order language with variables ranging over such functions, symbols for 0, +, −, min , max , and absolute value, and a ternary relation f = g + O(h). We show that the set of quantifier-free formulas in this language that are valid in the intended class of interpretations is decidable and does not depend on the underlying set, S, or the ordered ring, R. If R is a subfield of the real numbers, we can add a constant 1 function, as well as multiplication by constants from any computable subfield. We obtain further decidability results for certain situations in which one adds symbols denoting the elements of a fixed sequence of functions of strictly increasing rates of growth.     

Improved algorithms for finding length-bounded two vertex-disjoint paths in a planar graph and minmax k vertex-disjoint paths in a directed acyclic graph

This paper is composed of two parts. In the first part, an improved algorithm is presented for the problem of finding length-bounded two vertex-disjoint paths in an undirected planar graph. The presented algorithm requires O(n³bmin) time and O(n²bmin) space, where bmin is the smaller of the two given length bounds. In the second part of this paper, we consider the minmax k vertex-disjoint paths problem on a directed acyclic graph, where k?2 is a constant. An improved algorithm and a faster approximation scheme are presented. The presented algorithm requires O(nk+1Mk−1) time and O(nkMk−1) space, and the presented approximation scheme requires O((1/?)^k−1n2klogk−1M) time and O((1/?)^k−1n2k−1logk−1M) space, where ? is the given approximation parameter and M is the length of the longest path in an optimal solution.     

On circular-L(2, 1)-labellings of products of graphs

Let m, j and k be positive integers. An m-circular-L(j, k)-labelling of a graph G is an assignment f from { 0, 1,?…?, m?1} to the vertices of G such that, for any two vertices u and v, |f(u)?f(v)|_m≥j if uv∈E(G), and |f(u)?f(v)|_m≥k if d_G(u, v)=2, where |a|_m=min{a, m?a}. The minimum m such that G has an m-circular-L(j, k)-labelling is called the circular-L(j, k)-labelling number of G. This paper determines the circular-L(2, 1)-labelling numbers of the direct product of a path and a complete graph and of the Cartesian product of a path and a cycle.     

The Zariski topology on the spectrum of prime <Emphasis Type="Italic">L</Emphasis>-submodules

Let R be a commutative ring with identity and let M be an R-module. We topologize L − Spec(M), the collection of all prime L-submodules of M, analogous to that for FSpec(R), the spectrum of fuzzy prime ideals of R, and investigate the properties of this topological space. In particular, we will study the relationship between L − Spec(M) and L − Spec(R/Ann(M)) and obtain some results.     

Computing the local metric dimension of a graph from the local metric dimension of primary subgraphs

For an ordered subset W= w₁, w₂,?…?w_k of vertices and a vertex u in a connected graph G, the representation of u with respect to W is the ordered k-tuple r(u|W)=(d(u, w₁), d(u, w₂),?…?, d(u, w_k)), where d(x, y) represents the distance between the vertices x and y. The set W is a local metric generator for G if every two adjacent vertices of G have distinct representations. A minimum local metric generator is called a local metric basis for G and its cardinality the local metric dimension of G. We show that the computation of the local metric dimension of a graph with cut vertices is reduced to the computation of the local metric dimension of the so-called primary subgraphs. The main results are applied to specific constructions including bouquets of graphs, rooted product graphs, corona product graphs, block graphs and chain of graphs.     

Algorithms for (0, 1, d)-graphs with d constrains

Let G be a graph with vertex set V(G). Let n, k, d be non-negative integers such that n+2k+d≤|V(G)|?2 and |V(G)|?n?d are even. A matching which saturates exactly |V(G)|?d vertices is called a defect-d matching of G. If when deleting any n vertices the remaining subgraph contains a matching of k edges and every k-matching can be extended to a defect-d matching, then G is said to be an (n, k, d)-graph. We present an algorithm to determine (0, 1, d)-graphs with d constraints. Moreover, we solve the problem of augmenting a bipartite graph G=(B, W) to be a (0, 1, d)-graph by adding fewest edges, where d=∥B|?|W∥. The latter problem is applicable to the job assignment problem, where the number of jobs does not equal the number of persons.     

Weighted Mean of a Pair of Graphs

G and G′, with d(G, G′) being the edit distance of G and G′, the weighted mean of G and G′ is a graph G″ that has edit distances d(G, G″) and d(G″, G′) to G and G′, respectively, such that d(G, G″) + d(G″, G′) = d(G,G′). We'll show formal properties of the weighted mean, describe a procedure for its computation, and give examples. Received April 9, 2001     

On Minimum-Area Hulls

We study some minimum-area hull problems that generalize the notion of convex hull to star-shaped and monotone hulls. Specifically, we consider the minimum-area star-shaped hull problem: Given an n -vertex simple polygon P , find a minimum-area, star-shaped polygon P ^* containing P . This problem arises in lattice packings of translates of multiple, nonidentical shapes in material layout problems (e.g., in clothing manufacture), and has been recently posed by Daniels and Milenkovic. We consider two versions of the problem: the restricted version, in which the vertices of P ^* are constrained to be vertices of P , and the unrestricted version, in which the vertices of P ^* can be anywhere in the plane. We prove that the restricted problem falls in the class of ``3sum-hard' (sometimes called ``n ² -hard') problems, which are suspected to admit no solutions in o(n ² ) time. Further, we give an O(n ² ) time algorithm, improving the previous bound of O(n ⁵ ) . We also show that the unrestricted problem can be solved in O(n ² p(n)) time, where p(n) is the time needed to find the roots of two equations in two unknowns, each a polynomial of degree O(n) . We also consider the case in which P ^* is required to be monotone, with respect to an unspecified direction; we refer to this as the minimum-area monotone hull problem. We give a matching lower and upper bound of Θ(n log n) time for computing P ^* in the restricted version, and an upper bound of O(n q(n)) time in the unrestricted version, where q(n) is the time needed to find the roots of two polynomial equations in two unknowns with degrees 2 and O(n) . Received November 1996; revised March 1997.     

On the locatic number of graphs

A vertex v of a connected graph G distinguishes a pair u, w of vertices of G if d(v, u)≠d(v, w), where d(·,·) denotes the length of a shortest path between two vertices in G. A k-partition Π={S ₁, S ₂, …, S _k } of the vertex set of G is said to be a locatic partition if for every pair of distinct vertices v and w of G, there exists a vertex s∈S _i for all 1≤i≤k that distinguishes v and w. The cardinality of a largest locatic partition is called the locatic number of G. In this paper, we study the locatic number of paths, cycles and characterize all the connected graphs of order n having locatic number n, n?1 and n?2. Some realizable results are also given in this paper.     

Improved upper bounds on the L(2,1) -labeling of the skew and converse skew product graphs

An L(2,1)-labeling of a graph G is a function f from the vertex set V(G) to the set of all nonnegative integers such that |f(x)−f(y)|≥2 if d(x,y)=1 and |f(x)−f(y)|≥1 if d(x,y)=2, where d(x,y) denotes the distance between x and y in G. The L(2,1)-labeling number λ(G) of G is the smallest number k such that G has an L(2,1)-labeling with max{f(v):vV(G)}=k. Griggs and Yeh conjecture that λ(G)≤Δ² for any simple graph with maximum degree Δ≥2. This paper considers the graph formed by the skew product and the converse skew product of two graphs with a new approach on the analysis of adjacency matrices of the graphs as in [W.C. Shiu, Z. Shao, K.K. Poon, D. Zhang, A new approach to the L(2,1)-labeling of some products of graphs, IEEE Trans. Circuits Syst. II: Express Briefs (to appear)] and improves the previous upper bounds significantly.     

A cubically convergent iteration method for multiple roots of f(x)=0

This paper describes a cubically convergent iteration method for finding the multiple roots of nonlinear equations, f(x)=0, where f:?→? is a continuous function. This work is the extension of our earlier work [P.K. Parida, and D.K. Gupta, An improved regula-falsi method for enclosing simple zeros of nonlinear equations, Appl. Math. Comput. 177 (2006), pp. 769–776] where we have developed a cubically convergent improved regula-falsi method for finding simple roots of f(x)=0. First, by using some suitable transformation, the given function f(x) with multiple roots is transformed to F(x) with simple roots. Then, starting with an initial point x ₀ near the simple root x* of F(x)=0, the sequence of iterates {x _n }, n=0, 1, … and the sequence of intervals {[a _n , b _n ]}, with x*∈{[a _n , b _n ]} for all n are generated such that the sequences {(x _n ?x*)} and {(b _n ?a _n )} converges cubically to 0 simultaneously. The convergence theorems are established for the described method. The method is tested on a number of numerical examples and the results obtained are compared with those obtained by King [R.F. King, A secant method for multiple roots, BIT 17 (1977), pp. 321–328.].     

Simulating parallel neighboring communications among square meshes and square toruses

Given a d-dimensional square mesh or square torus G and a c-dimensional square mesh or square torus H such that G and H are of the same size but may differ in dimensions and shapes, we study the problem of simulating in H parallel neighboring communications in G. We assume that the nodes in H have only unit-size buffers associated with the links, and that packets can be sent and received simultaneously from all outbound links and inbound links of the nodes. For permutation-type parallel neighboring communications, for all the combinations of graph types and graph shapes of G and H, except for the case in which d < c and c is not divisible by d, we show that H can simulate G either optimally or optimally up to a constant multiplicative factor for fixed values of d and c. For scatter-type parallel neighboring communications, for some special cases of G and H, we also show that H can optimally simulate G. All these simulation times are smaller than the diameter of H, the lower bound on the routing complexity to support general data permutations in H.This work has been partially supported by National Science Foundation PYI award DCR84-51408, IBM research grant, AT&T Information System research grant, National Science Foundation CER grant MCS82-19196, Army Research Office grant DAAG-29-84-K-0061, Canada NSERC research grant OGP0041648, and British Science and Engineering Research Council visiting fellowship research grant.     

On Nonnegative matrices generating a finite multiplicative monoid

Square nonnegative matrices with the property that the multiplicative monoid M(A) generated by the matrix A is finite are characterized in several ways. At first, the least general upper bound for the cardinality of M(A) is derived. Then it is shown that any square nonnegative matrix is cogredient to a lower triangular block form with the diagonal consisting of three blocks L, A ₀, and M where L and M are strictly lower triangular, A ₀ has no zero rows or columns, and M(A) is finite if and only if. M(A ₀) is so. Several criteria for, M(A ₀) to be finite are presented. One of the normal forms of A applies very well to the characterization of the nonnegative solutions of each of the matrix equations X ^k = 0, X ^k = 1, X ^k = X, and X ^k = X ^T where k > 1 is an integer. It also leads to a polynomial time algorithm for deciding whether or not M(A) is finite, if the entries of A are nonnegative rationals.     

NP对PP

主要目的是研究NP与PP的关系。引入了一个NP的等价的随机定义。基于此等价定义,定义了另一个随机复杂性类：SUPER－NP。虽然SUPER－NP与NP非常接近,但令人吃惊的是发现了PP包含于SUPER－NP,从而NP包含于PP包含于SUPER－NP。考虑到NP＝PCP（log,O(1))以及NP和SUPER－NP的相似性,也希望能通过证明SUPER－NP包含于PCP（log^2,O(1))来解决PP包含于PCP（log^2,O(1))的猜想。     

On the 1-fault hamiltonicity for graphs satisfying Ore's theorem and its generalization

Consider any undirected and simple graph G=(V, E), where V and E denote the vertex set and the edge set of G, respectively. Let |G|=|V|=n. The well-known Ore's theorem states that if deg_G(u)+deg_G(v)≥n+k holds for each pair of nonadjacent vertices u and v of G, then G is traceable for k=?1, hamiltonian for k=0, and hamiltonian-connected for k=1. Lin et al. generalized Ore's theorem and showed that under the same condition as above, G is r*-connected for 1≤r≤k+2 with k≥1. In this paper, we improve both theorems by showing that the hamiltonicity or r*-connectivity of any graph G satisfying the condition deg_G(u)+deg_G(v)≥n+k with k≥?1 is preserved even after one vertex or one edge is removed, unless G belongs to two exceptional families.