Graph Decomposition

A graceful labeling of a graph G with n edges is an injection f : V (G) →{ 0,1,2,···,n} with the property that the resulting edge labels are also distinct, where an edge incident with vertices u and v is assigned the label |f(u)−f(v)|. A graph which admits a graceful labeling is called a graceful graph. In this paper, inspired by Koh [9] method, which combines a known graceful trees to obtain a larger graceful trees, we introduced a new method of combining graceful trees called recursive attachment method, and we show that the recursively attached tree Ti = Ti−1 ⊕TAi−1 is graceful, for i ≥ 1, where T0 is a base tree which is taken as a caterpillar and TAi−1 is an attachment tree which taken as any caterpillar. Here Ti−1 ⊕TAi−1 represents a tree obtained by attaching a copy of TAi−1 at each vertex of degree at least two in Ti−1, for i ≥ 1. Consequently the graceful tree conjecture is true for every recursively attached caterpillar tree Ti, fori ≥ 1.

A labeling (or valuation) of a graph G is an assignment of integers to the vertices of G subject to certain conditions. A hierarchy of graph labelings was introduced by Rosa in the late 1960s. Rosa showed that certain basic labelings of a graph G with n edges yielded cyclic G-decompositions of K 2n+1 while other stricter labelings yielded cyclic G-decompositions of K 2nx+1 for all natural numbers x. Rosa-type labelings are labelings with applications to cyclic graph decompositions. We survey various Rosa-type labelings and summarize some of the related results.

Evolutionary algorithms (EAs) have been widely used in handling various water resource optimization problems in recent years. However, it is still challenging for EAs to identify near-optimal solutions for realistic problems within the available computational budgets. This paper introduces a novel multiobjective optimization method to improve the efficiency of a typically difficult water resource problem: water distribution network (WDN) design. In the proposed approach, a WDN is decomposed into different sub-networks using decomposition techniques. EAs optimize these sub-networks individually, generating Pareto fronts for each sub-network with great efficiency. A propagation method is proposed to evolve Pareto fronts of the sub-networks towards the Pareto front for the full network while eliminating the need to hydraulically simulate the intact network itself. Results from two complex realistic WDNs show that the proposed approach is able to find better fronts than conventional full-search algorithms (optimize the entire network without decomposition) with dramatically improved efficiency.

During the last three decades, different types of decompositions have been pro- cessed in thefield of graph theory. Among these we mention: decompositions based on the additivity of some characteristics of the graph, decompositions where the adjacency law be- tween the subsets of the partition is known, decompositions where the subgraph induced by every subset of the partition must have

Development of robust dynamical systems and networks such as autonomous aircraft systems capable of accomplishing complex missions faces challenges due to the dynamically evolving uncertainties coming from model uncertainties, necessity to operate in a hostile cluttered urban environment, and the distributed and dynamic nature of the communication and computation resources. Model-based robust design is difficult because of the complexity of

System-level modeling, simulation, and synthesis using electronic design automation (EDA) tools are key steps in the design process for communication and signal processing systems, and the synchronous dataflow (SDF) model of computation is widely used in EDA tools for these purposes. Behavioral representations of modern wireless communication systems typically result in critical SDF graphs: These consist of hundreds of components (or more) and involve complex intercomponent connections with highly multirate relationships (i.e., with large variations in average rates of data transfer or component execution across different subsystems). Simulating such systems using conventional SDF scheduling techniques generally leads to unacceptable simulation time and memory requirements on modern workstations and high-end PCs. In this article, we present a novel simulation-oriented scheduler (SOS) that strategically integrates several techniques for graph decomposition and SDF scheduling to provi...

Development of robust dynamical systems and networks such as autonomous aircraft systems capable of accomplishing complex missions faces challenges due to the dynamically evolving uncertainties coming from model uncertainties, necessity to operate in a hostile cluttered urban environment, and the distributed and dynamic nature of the communication and computation resources. Model-based robust design is difficult because of the complexity of

Traditional intrusion detection systems (IDSs) focus on low-level attacks or anomalies, and raise alerts independently, though there may be logical connections between them. In situations where there are intensive intrusions, not only will actual alerts be mixed with false alerts, but the amount of alerts will also become unmanageable. As a result, it is difficult for human users or intrusion response systems to understand the alerts and take appropriate actions. Several complementary alert correlation methods have been proposed to address this problem. As one of these methods, we have developed a framework to correlate intrusion alerts using prerequisites of intrusions. In this paper, we continue this work to study the feasibility of this method in analyzing real-world, intensive intrusions. In particular, we develop three utilities (called adjustable graph reduction, focused analysis, and graph decomposition) to facilitate the analysis of large sets of correlated alerts. We study the effectiveness of the alert correlation method and these utilities through a case study with the network traffic captured at the DEF CON 8 Capture the Flag (CTF) event. Our results show that these utilities can simplify the analysis of large amounts of alerts, and also reveals several attack strategies that were repeatedly used in the DEF CON 8 CTF event.

This paper explores the embeddings of multidimensional meshes into minimal Boolean cubes by graph decomposition. The dilation and the congestion of the product graph (G1 × G2) → (H1 × H2) is the maximum of the dilation and congestion for the two embeddings G1 → H1 and G2 → H2. The graph decomposition technique can be used to improve the average dilation and average congestion. The graph decomposition technique combined with some particular two-dimensional embeddings allows for minimal-expansion, dilation-two, congestion-two embeddings of about 87% of all two-dimensional meshes, with a significantly lower average dilation and congestion than by modified line compression. For three-dimensional meshes we show that the graph decomposition technique, together with two three-dimensional mesh embeddings presented in this paper and modified line compression, yields dilation-two embeddings of more than 96% of all three-dimensional meshes contained in a 512 × 512 × 512 mesh. The graph decomposition technique is also used to generalize the embeddings to meshes with wrap-around. The dilation increases by at most one compared to a mesh without wraparound. The expansion is preserved for the majority of meshes, if a wraparound feature is added to the mesh.

The question of finding feasible ways for estimating probability distributions is one of the main challenges for Estimation of Distribution Algorithms (EDAs). To estimate the distribution of the selected solutions, EDAs use factorizations constructed according to graphical models. The class of factorizations that can be obtained from these probability models is highly constrained. Expanding the class of factorizations that could be employed for probability approximation is a necessary step for the conception of more robust EDAs. In this paper we introduce a method for learning a more general class of probability factorizations. The method combines a reformulation of a probability approximation procedure known in statistical physics as the Kikuchi approximation of energy, with a novel approach for finding graph decompositions. We present the Markov Network Estimation of Distribution Algorithm (MN-EDA), an EDA that uses Kikuchi approximations to estimate the distribution, and Gibbs Sampling (GS) to generate new points. A systematic empirical evaluation of MN-EDA is done in comparison with different Bayesian network based EDAs. From our experiments we conclude that the algorithm can outperform other EDAs that use traditional methods of probability approximation in the optimization of functions with strong interactions among their variables.