15426263
Super connectivity of line graphs
The super connectivity κ' and the super edge-connectivity λ' are more refined network reliability indices than connectivity κ and edge-connectivity λ. This paper shows that for a connected graph G with order at least four rather than a star and its line graph L(G), κ'(L(G)) = λ'(G) if and only if G is not super-λ'. As a consequence, we obtain the result of Hellwig et al. [Note on the connectivity of line graphs, Inform. Process. Lett. 91 (2004) 7] that κ(L(G)) = λ'(G). Furthermore, the authors show that the line graph of a super-λ' graph is super-λ if the minimum degree is at least three.
872168
The super connectivity of shuffle-cubes
The shuffle-cube SQn, where n ≡ 2 (mod 4), a new variation of hypercubes proposed by Li et al. [T.-K. Li, J.J.M. Tan, L.-H. Hsu, T.-Y. Sung, The shuffle-cubes and their generalization, Inform. Process. Lett. 77 (2001) 35-41], is an n-regular n-connected graph. This paper determines that the super connectivity of SQn is 2n - 4 and the super edge-connectivity is 2n - 2 for n ≥ 6.
15417765
The proof of a conjecture of Bouabdallah and Sotteau
Let G be a connected graph of order n. A routing in G is a set of n(n 1) fixed paths for all ordered pairs of vertices of G. The edge-forwarding index of G, (G), is the minimum of the maximum number of paths specified by a routing passing through any edge of G taken over all routings in G, and ,n is the minimum of (G) taken over all graphs of order n with maximum degree at most . To determine n 2p 1,n for 4p 2p/3 1 ≤ n ≤ 6p, A. Bouabdallah and D. Sotteau proposed the following conjecture in [On the edge forwarding index problem for small graphs, Networks 23 (1993), 249–255]. The set 3 {1, 2, . . . , (4p)/3} can be partitioned into 2p pairs plus singletons such that the set of differences of the pairs is the set 2 {1, 2, . . . , p}. This article gives a proof of this conjecture and determines that n 2p 1,n is equal to 5 if 4p 2p/3 1 ≤ n ≤ 6p and to 8 if 3p p/3 1 ≤ n ≤ 3p (3p)/5 for any p ≥ 2. © 2004 Wiley Periodicals, Inc. NETWORKS, Vol. 44(4), 292–296 2004
29728094
Graph Bandwidth of Weighted Caterpillars
Graph bandwidth minimization (GBM) is a classical and challenging problem in graph algorithms and combinatorial optimization. Most of existing researches on this problem have focused on unweighted graphs. In this paper, we study the bandwidth minimization problem of weighted caterpillars, and propose several algorithms for solving various types of caterpillars and general graphs. More specifically, we show that the GBM problem on caterpillars with hair-length at most 2 and the GBM problem on star-shape caterpillars are NP-complete, and give a lower bound of the graph bandwidth for general weighted graphs. For caterpillars with hair-length at most 1, we present an O(n log n log(nwmax))-time algorithm to compute an optimal bandwidth layout, where n is the total number of vertices in the graph and wmax is the maximum wedge weight. For caterpillars with hair-length at most k, we give a k-approximation algorithm. For arbitrary caterpillars and general graphs, we give a heuristic algorithm and some experimental results. Experiments show that the solutions obtained by our heuristic algorithm are roughly within a factor of c log(n) of the lower bound for a small number c, which is consistent with the inapproximability results of this problem (i.e., no constant approximation for the GBM problem unless P = NP). © 2006 Published by Elsevier B.V.
14297439
Geometric permutations of high dimensional spheres
We prove the maximum number of geometric permutations, induced by line transversals to a set of <i>n</i> pairwise disjoint congruent spheres in <i>R<sup>d</sup></i> with <i>d</i> ⪈ 3, is no more than 4 when <i>n</i> is sufficiently large, achieving the best known upper bound for this problem. We also prove the maximum number of geometric permutations of a set of <i>n</i> noncongruent spheres of bounded radius ratio in R<sup>d</sup>, <i>d</i> ⪈ 3, is at most 2<sup>[√2<i>M</i>]+1</sup>, where <i>M</i> is the ratio or the largest radius and the smallest radius. Our result settles a conjecture in combinatorial geometry.
11486473
Circumference of Graphs with Bounded Degree
Karger, Motwani, and Ramkumar Algorithmica, 18 (1997), pp. 82--98] have shown that there is no constant approximation algorithm to find a longest cycle in a Hamiltonian graph, and they conjectured that this is the case even for graphs with bounded degree. On the other hand, Feder, Motwani, and Subi [SIAM J. Comput., 31 (2002), pp. 1596--1607] have shown that there is a polynomial time algorithm for finding a cycle of length $n^{\log_32}$ in a 3-connected cubic n-vertex graph. In this paper, we show that if G is a 3-connected n-vertex graph with maximum degree at most d, then one can find, in O(n3) time, a cycle in G of length at least $\Omega(n^{\log_b2})$, where $b=2(d-1)^2+1$.
41027997
Graph bandwidth of weighted caterpillars
Graph bandwidth minimization (GBM) is a classical and challenging problem in graph algorithms and combinatorial optimization. Most of existing researches on this problem have focused on unweighted graphs. In this paper, we study the bandwidth minimization problem of weighted caterpillars, and propose several algorithms for solving various types of caterpillars and general graphs. More specifically, we show that the GBM problem on caterpillars with hair-length at most 2 and the GBM problem on star-shape caterpillars are NP-complete, and give a lower bound of the graph bandwidth for general weighted graphs. For caterpillars with hair-length at most 1, we present an O(n log n log(nwmax))-time algorithm to compute an optimal bandwidth layout, where n is the total number of vertices in the graph and wmax is the maximum wedge weight. For caterpillars with hair-length at most k, we give a k-approximation algorithm. For arbitrary caterpillars and general graphs, we give a heuristic algorithm and some experimental results. Experiments show that the solutions obtained by our heuristic algorithm are roughly within a factor of clog(n) of the lower bound for a small number c, which is consistent with the inapproximability results of this problem (i.e., no constant approximation for the GBM problem unless P = NP).
2025275
Optimizing the sum of linear fractional functions and applications
1 I n t r o d u c t i o n . The problem of optimizing the sum of linear fractional functions (SOLF) is defined as follows: max f(Zl,...,Xd)= ~ [ : 7 : ~ d ) (Xl ..... z~t)ES i=1 ~ a r t m e n t of Computer Science and Engineering, University of Notre Dame, Notre Dame, IN 46556, USA, {chen,odaescu,xwu,jxu}@cse.nd.edu. This research was supported in part by the National Science Foundation under Grant CCR-9623585. tThe work of this author was supported in part by a fellowship of the Center for Applied Mathematics, University of Notre Dame, Notre Dame, Indiana, USA. t Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, 2-12-1 O-okayama, Meguro-ku, Tokyo 152, Japan, daiOis.titech.ac.jp. Grant-in-Aid of Ministry of Science, Culture and Education of Japan, No. 10780274. §Department of Architecture and Architectural Systems, Graduate School of Engineering, Kyoto University, Yoshida-Honmachi, Sakyo-ku, Kyoto, 606-8501 Japan, naoki~archi.kyoto-u.ac.jp. Partially supported by Grant-inAid for Scientific Research on Priority Areas (B) by the Ministry of Education, Science, Sports and Culture of Japan. IThe work of this author was supported in part by a fellowship of the Center for Applied Mathematics, and by a summer graduate research fellowship of the Graduate School, University of Notre Dame, Notre Dame, Indiana, USA. such that for each i = 1 , 2 , . . . , m , ni(Xl,...,Xd) and di(Xl,...,Xd) are linear functions in a fixed d-D space ]~d, di (x t , . . . ,Xd) ~ 0 for any (Xl , . . . ,Xd) E S, the feasible domain S is defined by n linear constraints (i.e., half-spaces in Rd), and S ~ ¢. Each linear fractional term ~(~1 ..... ~ ) ddxl,...,z~) is called a ratio. The SOLF problem appears in the algorithmic solutions for several geometric optimization problems. In layered manufacturing, Majhi et al. [17], [18] showed that the length-optimal supports for a simple nonconvex polygon can be obtained by solving the 1-D SOLF problem on an interval. Arkin et al. [1] formulated the problem of finding a minimum-area starshaped or monotone polygon that contains a simple polygon, which appears in material layout [1], [8], [21] (e.g., cloth manufacturing) and manufacturing [18], as one of optimizing the sum of 2-D fractional polynomials of degrees 3 and 2 under linear constraints. We will show that the objective function of this problem can actually be simplified to the 2-D SOLF form. For the problem of finding optimal penetrations among weighted regions [5], we shall show tha t the L1 and Leo versions of this problem have the 2-D SOLF problem as a key subproblem. The SOLF problem also arises in other areas. For example, in combinatorial optimization, a class of problems on finding a structure with the maximum (or minimum) mean-weight cost can be solved by optimizing some linear fractional functions [23]. In operations research, it was shown in [11], [24] that many economic applications (e.g., maximization of productivity, return on investment, and return/risk) can be reduced to solving the SOLF problem. Quite a few solutions have been given for the SOLF problem [1], [11], [15], [17], [18], [25]. It is interesting to note that the SOLF related problems were originally studied in economic applications, where the number of variables is usually much bigger than the number of fractional terms. As a result, many previous SOLF algorithms were designed to target problems with only a few fractional terms (less than 10) for a reasonable