7639532
Fault diameter of Cartesian product graphs
The (k-1)-fault diameter D"k(G) of a k-connected graph G is the maximum diameter of G-F for any F@?V(G) with |F|=1 for i=1,2.
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.
297940
Connectivity of Cartesian product graphs
Use v"i,@k"i,@l"i,@d"i to denote order, connectivity, edge-connectivity and minimum degree of a graph G"i for i=1,2, respectively. For the connectivity and the edge-connectivity of the Cartesian product graph, up to now, the best results are @k(G"1xG"2)>=@k"1+@k"2 and @l(G"1xG"2)>=@l"1+@l"2. This paper improves these results by proving that @k(G"1xG"2)>=min{@k"1+@d"2,@k"2+@d"1} and @l(G"1xG"2)=min{@d"1+@d"2,@l"1v"2,@l"2v"1} if G"1 and G"2 are connected undirected graphs; @k(G"1xG"2)>=min{@k"1+@d"2,@k"2+@d"1,2@k"1+@k"2,2@k"2+@k"1} if G"1 and G"2 are strongly connected digraphs. These results are also generalized to the Cartesian products of n(>=3) connected graphs and n strongly connected digraphs, respectively.
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.
39336188
Local Strongly Arc-Connectivity in Regular Bipartite Digraphs
We show that for any vertex x of a d-regular bipartite digraph there are a vertex y, in the other class of the bipartition, and d (x, y)-paths and d (y, x)-paths such that all 2d of them are pairwise arc-disjoint. This result generalizes a theorem of Hamidoune and Las Vergnas for graphs.
30370540
On (d, 2)-dominating numbers of binary undirected de Bruijn graphs
Abstract In this paper, we show that: (i) For n-dimensional undirected binary de Bruijn graphs, UB(n), n⩾4, there is a vertex x=x 1 x 1 x 1 ⋯ x 1 x 1 (x1=1 or 0) such that for any other vertex t there exist at least two internally disjoint paths of length at most n−1 between x and t in UB(n), i.e., the (n−1,2)-dominating number of UB(n) is equal to one. (ii) For n⩾5, let S={100⋯01,011⋯10}. For any other vertex t there exist at least two internally disjoint paths of length at most n−2 between t and S in UB(n), i.e., the (n−2,2)-dominating number of UB(n) is no more than 2.
5147011
Wide Diameters of Cartesian Product Graphs and Digraphs
In graph theory and study of fault tolerance and transmission delay of networks, connectivity and diameter of a graph are two very important parameters and have been deeply studied by many authors. Wide diameter combining connectivity with diameter is a more important parameter to measure fault tolerance and efficiency of parallel processing computer networks and has received much attention in the recent years. Diameter with width k of a graph G is defined as the minimum integer d for which between any two distinct vertices in G there exist at least k internally disjoint paths of length at most d. In the present paper, the tight upper bounds of wide diameter of the Cartesian product graphs are obtained. Some known results can be deduced or improved from ours.
42016450
Geometric permutations of higher dimensional spheres
In this paper, we prove that the maximum number of geometric permutations (induced by line transversals) of a set of n pairwise disjoint spheres with a bounded radius ratio in Rd for d ≥ 3 is at most 2⌊√2M⌋+1, where M is the ratio of the largest radius and the smallest radius of the spheres. Setting M to 1, this gives an upper bound of 4 on the maximum number of geometric permutations for congruent spheres in Rd, matching a recently independently discovered result [Y. Zhou, S. Suri, in: Proc. of 12th Annual ACM-SIAM Symp. on Discrete Algorithms, 2001, pp. 234-243] on this case. Our result settles a conjecture in combinatorial geometry.
757091
Efficient Job Scheduling Algorithms with Multi-type Contentions
In this paper, we consider an interesting generalization of the classic job scheduling problem in which each job needs to compete for not only machines but also other types of resources The contentions among jobs for machines and resources could interfere with each other, which complicates the problem dramatically We present a family of approximation algorithms for solving several variants of the problem by using a generic algorithmic framework Our algorithms achieve a constant approximation ratio (i.e., 3) if there is only one type of resources or certain dependency relation exists among multiple types of resources For the case that r unrelated resources are given, the approximation ratio of our algorithm becomes k + 2, where k ≤ r is a constant depending on the problem instance As an application, we also show that our techniques can be easily applied to optical burst switching (OBS) networks for deriving more efficient wavelength scheduling algorithms.
8376312
Transparent runtime randomization for security
A large class of security attacks exploit software implementation vulnerabilities such as unchecked buffers. This paper proposes transparent runtime randomization (TRR), a generalized approach for protecting against a wide range of security attacks. TRR dynamically and randomly relocates a program's stack, heap, shared libraries, and parts of its runtime control data structures inside the application memory address space. Making a program's memory layout different each time it runs foils the attacker's assumptions about the memory layout of the vulnerable program and makes the determination of critical address values difficult if not impossible. TRR is implemented by changing the Linux dynamic program loader, hence it is transparent to applications. We demonstrate that TRR is effective in defeating real security attacks, including malloc-based heap overflow, integer overflow, and double-free attacks, for which effective prevention mechanisms are yet to emerge. Furthermore, TRR incurs less than 9% program startup overhead and no runtime overhead.
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 $b=2(d-1)^2+1$.