Fuzzy Chordal Graphs and Its Properties

Some properties of edge intersection graphs of single-bend paths on a grid

In this chapter, we give another characterisation of planar graphs, due to Chernyak (1980). In fact, we shall prove a general theorem about consistently orientable bonds in an arbitrary graph. The characterisation of planar graphs is obtained by dualising this theorem. The argument depends on a property of chordal graphs. Accordingly, chordal graphs are defined and studied in the next section.

Finding a central vertex in an HHD-free graph

Treewidth for graphs with small chordality

An important property of graph which concerns their applications to networks is Hamiltonicity. Determining if a graph is hamiltonian is a NP-complete problem and no satisfactory characterization exists. Nevertheless, many sufficient conditions were found, usually expressed in terms of degree, connectivity, density, toughness, independent set, regularity and forbidden subgraphs. In 1973, Goodman and Hedetniemi gave two alternative sufficient conditions uniquely based on the possibility to decompose in some ways the graph into cliques. We first generalise one of these two conditions. We give a new clique decomposition condition which proves the hamiltonicity of a broader class of graphs. Then, we present a polynomial algorithm which decides the existence of such a decomposition for simplicial-connected graphs. A graph is simplicial-connected if every vertex is connected by a walk to a simplicial one1. In particular, each connected graph containing a simplicial vertex is simplicial-connected, and so is every chordal graph.

This paper presents a new register assignment heuristic for procedures in SSA Form, whose interference graphs are chordal; the heuristic is called optimistic chordal coloring (OCC). Previous register assignment heuristics eliminate copy instructions via coalescing, in other words, merging nodes in the interference graph. Node merging, however, can not preserve the chordal graph property, making it unappealing for SSA-based register allocation. OCC is based on graph coloring, but does not employ coalescing, and, consequently, preserves graph chordality, and does not increase its chromatic number; in this sense, OCC is conservative as well as optimistic. OCC is observed to eliminate at least as many dynamically executed copy instructions as iterated register coalescing (IRC) for a set of chordal interference graphs generated from several Mediabench and MiBench applications. In many cases, OCC and IRC were able to find optimal or near-optimal solutions for these graphs. OCC ran 1.89x faster than IRC, on average.

In this paper we show that every chordal graph with n vertices and m edges admits an additive 4-spanner with at most 2n--2 edges and an additive 3-spanner with at most O(n ċ log n) edges. This significantly improves results of Peleg and Schaffer from [Graph Spanners, J. Graph Theory, 13(1989), 99-116]. Our spanners are additive and easier to construct. An additive 4-spanner can be constructed in linear time while an additive 3-spanner is constructable in O(m ċ log n) time. Furthermore, our method can be extended to graphs with largest induced cycles of length k. Any such graph admits an additive (k + 1)-spanner with at most 2n - 2 edges which is constructable in O(n ċ k + m) time.

A characterization of signed graphs with generalized perfect elimination orderings

The concept of Fibonacci graphs introduced and developed by this author is critically reviewed. The concept has been shown to provide an easypencil-and-paper method of calculating characteristic, matching, counting, sextet, rook, color and king polynomials of graphs of quite large size with limited connectivities. For example, the coefficients of the matching polynomial of 18-annuleno—18-annulene can be obtainedby hand using the definition of Fibonacci graphs. They are (in absolute magnitudes): 1, 35, 557, 5337, 34 361, 157 081, 525 296, 1304 426, 2 416 571, 3 327 037, 3 362 528, 2 440 842, 1 229 614, 407 814, 81936, 8652, 361, 3. The theory of Fibonacci graphs is reviewed in an easy and detailed language. The theory leads to modulation of the polynomial of a graph with the polynomial of a path.

Based on definition and basic properties of rough graph, this paper gives different kinds of rough graph, which are weighted rough graph and directed rough graph. By using representation form of classical rough graph, representation forms of these rough graphs are also given, which can help us to research rough problems conveniently.

High throughput issue is one of the major obstacles for opening up the new big data stream computing era. A high throughput big data stream computing system is need, optimising the data flow graph is an important way to offer a high throughput computing environment in big data stream computing system. In this paper, the definition of data flow graph in big data stream computing is given and the properties of data flow graph are systematically analysed by referring to the stream computing theories. A series of optimisation strategies for data flow graph in big data stream computing environments are put forward, all those strategies will greatly improve the structure of data flow graph, and provide a high throughput environment for big data stream computing.

With the explosion of graph scale of social networks, it becomes increasingly impractical to study the original large graph directly. Being able to derive a representative sample of the original graph, graph sampling provides an efficient solution for social network analysis. We expect this sample could preserve some important graph properties and represent the original graph well. If one algorithm relies on the preserved properties, we can expect that it gives similar output on the original graph and the sampled graph. This leads to a systematic way to accelerate a class of graph algorithms. Our work is based on the idea of stratified sampling [14], a widely used technique in statistics. We propose a heuristic approach to achieve efficient graph sampling based on community structure of social networks. With the aid of ground-truth of communities available in social networks, we find out that sampling from communities preserves community- related graph properties very well. The experimental results show that our framework improves the performance of traditional graph sampling algorithms and therefore, is an effective method of graph sampling.

Network forensics focuses on the identification and investigation of internal and external network attacks, the reverse engineering of network protocols, and the uninstrumented investigation of networked devices. It lies at the intersection of digital forensics, incident response and network security. Network attacks exploit software and hardware vulnerabilities and communication protocols. The scope of a network forensic investigation can range from Internet-wide down to a single device’s network traffic. Network analysis tools (NATs) aid security professionals and law enforcement in the capturing, identification and analysis of network traffic. However, in most instances, the sheer volume of data to be analyzed is enormous and, despite some built-in NAT automation, the investigation of network traffic is often an arduous process. Furthermore, significant expert time remains wasted in the investigation of a high frequency of false positive alerting from automated systems. To address this globally impacting problem, artificial intelligence based approaches are becoming increasingly employed to automatically detect attacks and increase network traffic classification accuracy. This paper provides a comprehensive survey of the state-of-the-art in network forensics and the application of expert systems, machine learning, deep learning, and ensemble/hybrid approaches to a range of application areas in the field. These include network traffic analysis, intrusion detection systems, Internet-of-Things devices, cloud forensics, DNS tunneling, smart grid forensics, and vehicle forensics. In addition, the current challenges and future research directions for each of the aforementioned application areas is discussed.

Due to the limited capacity of GPU memory, the majority of prior work on graph applications on GPUs has been restricted to graphs of modest sizes that fit in memory. Recent hardware and software advances make it possible to address much larger host memory transparently as a part of a feature known as unified virtual memory. While accessing host memory over an interconnect is understandably slower, the problem space has not been sufficiently explored in the context of a challenging workload with low computational intensity and an irregular data access pattern such as graph traversal. We analyse the performance of breadth first search (BFS) for several large graphs in the context of unified memory and identify the key factors that contribute to slowdowns. Next, we propose a lightweight offline graph reordering algorithm, HALO (Harmonic Locality Ordering), that can be used as a pre-processing step for static graphs. HALO yields speedups of 1.5x-1.9x over baseline in subsequent traversals. Our method specifically aims to cover large directed real world graphs in addition to undirected graphs whereas prior methods only account for the latter. Additionally, we demonstrate ties between the locality ordering problem and graph compression and show that prior methods from graph compression such as recursive graph bisection can be suitably adapted to this problem.

Studies on periodic porosity and related properties and functions have been limited to insoluble solid-state materials. Self-assembly provides a straightforward and efficient strategy for the construction of soluble periodic porous supramolecular organic frameworks (SOFs) in water from rationally designed molecular building blocks. From rigid tri- and tetra-armed building blocks and cucurbitu[8]ril (CB[8]), a number of two-dimensional (2D) honeycomb, square and rhombic SOFs can be generated, which is driven by CB[8]-encapsulation-enhanced dimerization of two aromatic units on the periphery of the multi-armed molecules. By utilizing the same three-component host−guest motif as the driving force, three-dimensional (3D) diamondoid and cubic SOFs can be obtained from tetrahedral and [Ru(bipy)3]2+-derived octahedral monomers and CB[8]. All of the 2D and 3D periodic frameworks are soluble in water, and are able to maintain the periodicity as well as the pore sizes in the solid state. 3D SOFs are highly efficient homogeneous polycationic frameworks for reversible adsorption of anionic species including organic dyes, peptides, nucleic acids, drugs, dendrimers and Wells-Dawson-typed polyoxametallates (WD-POMs). WD-POM molecules adsorbed in the [Ru(bipy)3]2+-based SOF can catalyse the reduction of proton to H2 upon visible-light sensitization of [Ru(bipy)3]2+, which allows multiple electron transfer from [Ru(bipy)3]2+ to WD-POM. This review summarizes the design, formation and characterization of this new family of self-assembled frameworks, highlights their applications as homogeneous porous materials and finally outlines some future research directions.