32836256
Minimum average distance of strong orientations of graphs
The average distance of a graph (strong digraph) G, denoted by µ(G) is the average, among the distances between all pairs (ordered pairs) of vertices of G. If G is a 2-edge-connected graph, then µ→min(G) is the minimum average distance taken over all strong orientations of G. A lower bound for µ→min(G) in terms of the order, size, girth and average distance of G is established and shown to be sharp for several complete multipartite graphs. It is shown that there is no upper bound for µ→min(G) in terms of µ(G). However, if every edge of G lies on 3-cycle, then it is shown that µ→min(G) ≤ 7/4 µ(G). This bound is improved for maximal planar graphs to 5/3 µ(G) and even further to 3/2 µ(G) for eulerian maximal planar graphs and for outerplanar graphs with the property that every edge lies on 3-cycle. In the last case the bound is shown to be sharp.
45978330
Two sufficient conditions for a graph to be type 1
204989501
On clique partitions of split graphs
Abstract Split graphs are graphs formed by taking a complete graph and an empty graph disjoint from it and some or all of the possible edges joining the two. We prove that the problem of deciding the clique partition number is NP-complete, even when restricted to the class of split graphs.
41551452
Fault-Tolerant Adaptive and Minimal Routing in Mesh-Connected Multicomputers Using Extended Safety Levels
The minimal routing problem in mesh-connected multicomputers with faulty blocks is studied. Two-dimensional meshes are used to illustrate the approach. A sufficient condition for minimal routing in 2D meshes with faulty blocks is proposed. Unlike many traditional models that assume all the nodes know global fault distribution, our approach is based on the concept of an extended safety level, which is a special form of limited fault information. The extended safety level information is captured by a vector associated with each node. When the safety level of a node reaches a certain level (or meets certain conditions), a minimal path exists from this node to any nonfaulty nodes in 2D meshes. Specifically, we study the existence of minimal paths at a given source node, limited distribution of fault information, and minimal routing itself. We propose three fault-tolerant minimal routing algorithms which are adaptive to allow all messages to use any minimal path. We also provide some general ideas to extend our approaches to other low-dimensional mesh-connected multicomputers such as 2D tori and 3D meshes. Our approach is the first attempt to address adaptive and minimal routing in 2D meshes with faulty blocks using limited fault information.
1444873
A new fault information model for fault-tolerant adaptive and minimal routing in 3-D meshes
In this paper we rewrite Wang's Minimal-Connected-Component (MCC) model in 2D meshes without using global information so that not only the existence of a minimal path can be ensured at the source, but also such a path can be formed by routing decisions at intermediate nodes along the path. We extend this MCC model and the corresponding routing in 2D meshes to 3D meshes. It is based on our early work on fault tolerant adaptive and minimal routing and the boundary information model in 3D meshes. We study fault tolerant adaptive and minimal routing from the source and the destination and consider the positions of the source and destination when the new faulty components are constructed. Specifically, all faulty nodes will be contained in some disjoint faulty components and a healthy node will be included in a faulty component only if using it in the routing will definitely cause a non-minimal routing path. A sufficient and necessary condition is proposed for the existence of the minimal routing path in the presence of our faulty components. Based on such a condition, the corresponding routing will guarantee a minimal path whenever it exists.
6077474
Multicasting in injured hypercubes using limited global information
We study the multicast problem in injured hypercubes with faulty nodes by assuming that each node in a cube has limited global information, which is captured by the safety level associated with the node. Basically, the safety level is an approximate measure of the distribution of faulty nodes in the neighborhood. We first propose a safety-level based multicasting scheme which guarantees time optimality. Then an address-sum based multicasting scheme is studied which utilizes both the distribution of destination nodes and safety levels of neighboring nodes. Simulation results show that the traffic generated in both schemes is very close to the optimal solution. Time optimality is guaranteed when the source is safe and an additional time step is required when the source is unsafe in an n-cube with up to n - 1 node faults.<<ETX>>
5787542
Relaxed game chromatic number of trees and outerplanar graphs
Abstract This paper studies the relaxed game chromatic number of trees and outerplanar graphs. It is proved that if G is a tree then χ g d ( G )⩽2 for d ⩾2. If G is an outerplanar graph, then χ g d ( G )⩽5 for d ⩾2, and χ g d ( G )⩾3 for d ⩽4.