Studies on some graph operators and related topics

Department of Mathematics, Cochin University of Science and Technology

Studies on Some Graph Classes

Department of Mathematics, Cochin University of Science and Technology

Studies on some generalizations of line graph and the power domination problem in graphs

Department of Mathematics, Cochin University of Science and Technology

Clique Irreducibility and Clique Vertex Irreducibility of Graphs

A graphs G is clique irreducible if every clique in G of size at least two,has an edge which does not lie in any other clique of G and is clique reducible if it is not clique irreducible. A graph G is clique vertex irreducible if every clique in G has a vertex which does not lie in any other clique of G and clique vertex reducible if it is not clique vertex irreducible. The clique vertex irreducibility and clique irreducibility of graphs which are non-complete extended p-sums (NEPS) of two graphs are studied. We prove that if G(c) has at least two non-trivial components then G is clique vertex reducible and if it has at least three non-trivial components then G is clique reducible. The cographs and the distance hereditary graphs which are clique vertex irreducible and clique irreducible are also recursively characterized.

Convex extendable trees

The concept of convex extendability is introduced to answer the problem of finding the smallest distance convex simple graph containing a given tree. A problem of similar type with respect to minimal path convexity is also discussed.

Axiomatic Characterization of the Antimedian Function on Paths and Hypercubes

An antimedian of a pro le = (x1; x2; : : : ; xk) of vertices of a graph G is a vertex maximizing the sum of the distances to the elements of the pro le. The antimedian function is de ned on the set of all pro les on G and has as output the set of antimedians of a pro le. It is a typical location function for nding a location for an obnoxious facility. The `converse' of the antimedian function is the median function, where the distance sum is minimized. The median function is well studied. For instance it has been characterized axiomatically by three simple axioms on median graphs. The median function behaves nicely on many classes of graphs. In contrast the antimedian function does not have a nice behavior on most classes. So a nice axiomatic characterization may not be expected. In this paper such a characterization is obtained for the two classes of graphs on which the antimedian is well-behaved: paths and hypercubes.

Simultaneous Embeddings Of Graphs As Median And Antimedian Subgraphs

The distance DG(v) of a vertex v in an undirected graph G is the sum of the distances between v and all other vertices of G. The set of vertices in G with maximum (minimum) distance is the antimedian (median) set of a graph G. It is proved that for arbitrary graphs G and J and a positive integer r 2, there exists a connected graph H such that G is the antimedian and J the median subgraphs of H, respectively, and that dH(G, J) = r. When both G and J are connected, G and J can in addition be made convex subgraphs of H.

The Plurality Strategy on Graphs

The Majority Strategy for finding medians of a set of clients on a graph can be relaxed in the following way: if we are at v, then we move to a neighbor w if there are at least as many clients closer to w than to v (thus ignoring the clients at equal distance from v and w). The graphs on which this Plurality Strategy always finds the set of all medians are precisely those for which the set of medians induces always a connected subgraph

Median Sets and Median Number of a Graph

A profile is a finite sequence of vertices of a graph. The set of all vertices of the graph which minimises the sum of the distances to the vertices of the profile is the median of the profile. Any subset of the vertex set such that it is the median of some profile is called a median set. The number of median sets of a graph is defined to be the median number of the graph. In this paper, we identify the median sets of various classes of graphs such as Kp − e, Kp,q forP > 2, and wheel graph and so forth. The median numbers of these graphs and hypercubes are found out, and an upper bound for the median number of even cycles is established.We also express the median number of a product graph in terms of the median number of their factors.

On The Center Sets and Center Numbers of Some Graph Classes

For a set S of vertices and the vertex v in a connected graph G, max x2S d(x, v) is called the S-eccentricity of v in G. The set of vertices with minimum S-eccentricity is called the S-center of G. Any set A of vertices of G such that A is an S-center for some set S of vertices of G is called a center set. We identify the center sets of certain classes of graphs namely, Block graphs, Km,n, Kn −e, wheel graphs, odd cycles and symmetric even graphs and enumerate them for many of these graph classes. We also introduce the concept of center number which is defined as the number of distinct center sets of a graph and determine the center number of some graph classes

Consensus strategies for signed profiles on graphs

The median problem is a classical problem in Location Theory: one searches for a location that minimizes the average distance to the sites of the clients. This is for desired facilities as a distribution center for a set of warehouses. More recently, for obnoxious facilities, the antimedian was studied. Here one maximizes the average distance to the clients. In this paper the mixed case is studied. Clients are represented by a profile, which is a sequence of vertices with repetitions allowed. In a signed profile each element is provided with a sign from f+; g. Thus one can take into account whether the client prefers the facility (with a + sign) or rejects it (with a sign). The graphs for which all median sets, or all antimedian sets, are connected are characterized. Various consensus strategies for signed profiles are studied, amongst which Majority, Plurality and Scarcity. Hypercubes are the only graphs on which Majority produces the median set for all signed profiles. Finally, the antimedian sets are found by the Scarcity Strategy on e.g. Hamming graphs, Johnson graphs and halfcubes

Content aggregation on knowledge bases using graph clustering

Recently, research projects such as PADLR and SWAP have developed tools like Edutella or Bibster, which are targeted at establishing peer-to-peer knowledge management (P2PKM) systems. In such a system, it is necessary to obtain provide brief semantic descriptions of peers, so that routing algorithms or matchmaking processes can make decisions about which communities peers should belong to, or to which peers a given query should be forwarded. This paper proposes the use of graph clustering techniques on knowledge bases for that purpose. Using this clustering, we can show that our strategy requires up to 58% fewer queries than the baselines to yield full recall in a bibliographic P2PKM scenario.

Understanding Subsystems in Biology through Dimensionality Reduction, Graph Partitioning and Analytical Modeling

Biological systems exhibit rich and complex behavior through the orchestrated interplay of a large array of components. It is hypothesized that separable subsystems with some degree of functional autonomy exist; deciphering their independent behavior and functionality would greatly facilitate understanding the system as a whole. Discovering and analyzing such subsystems are hence pivotal problems in the quest to gain a quantitative understanding of complex biological systems. In this work, using approaches from machine learning, physics and graph theory, methods for the identification and analysis of such subsystems were developed. A novel methodology, based on a recent machine learning algorithm known as non-negative matrix factorization (NMF), was developed to discover such subsystems in a set of large-scale gene expression data. This set of subsystems was then used to predict functional relationships between genes, and this approach was shown to score significantly higher than conventional methods when benchmarking them against existing databases. Moreover, a mathematical treatment was developed to treat simple network subsystems based only on their topology (independent of particular parameter values). Application to a problem of experimental interest demonstrated the need for extentions to the conventional model to fully explain the experimental data. Finally, the notion of a subsystem was evaluated from a topological perspective. A number of different protein networks were examined to analyze their topological properties with respect to separability, seeking to find separable subsystems. These networks were shown to exhibit separability in a nonintuitive fashion, while the separable subsystems were of strong biological significance. It was demonstrated that the separability property found was not due to incomplete or biased data, but is likely to reflect biological structure.

Adaptive intelligent agent approach to guide the web navigation on the PLAN-G distance learning platform

Hypermedia systems based on the Web for open distance education are becoming increasingly popular as tools for user-driven access learning information. Adaptive hypermedia is a new direction in research within the area of user-adaptive systems, to increase its functionality by making it personalized [Eklu 961. This paper sketches a general agents architecture to include navigational adaptability and user-friendly processes which would guide and accompany the student during hislher learning on the PLAN-G hypermedia system (New Generation Telematics Platform to Support Open and Distance Learning), with the aid of computer networks and specifically WWW technology [Marz 98-1] [Marz 98-2]. The PLAN-G actual prototype is successfully used with some informatics courses (the current version has no agents yet). The propased multi-agent system, contains two different types of adaptive autonomous software agents: Personal Digital Agents {Interface), to interacl directly with the student when necessary; and Information Agents (Intermediaries), to filtrate and discover information to learn and to adapt navigation space to a specific student

TU Graz: Course: 707.000 Web Science and Web Technology: Lecture 3: Network Theory and Terminology

In this class, we will discuss network theory fundamentals, including concepts such as diameter, distance, clustering coefficient and others. We will also discuss different types of networks, such as scale-free networks, random networks etc. Readings: Graph structure in the Web, A. Broder and R. Kumar and F. Maghoul and P. Raghavan and S. Rajagopalan and R. Stata and A. Tomkins and J. Wiener Computer Networks 33 309--320 (2000) [Web link, Alternative Link] Optional: The Structure and Function of Complex Networks, M.E.J. Newman, SIAM Review 45 167--256 (2003) [Web link] Original course at: http://kmi.tugraz.at/staff/markus/courses/SS2008/707.000_web-science/