665 research outputs found
Complexity of Stability
Graph parameters such as the clique number, the chromatic number, and the independence number are central in many areas, ranging from computer networks to linguistics to computational neuroscience to social networks. In particular, the chromatic number of a graph (i.e., the smallest number of colors needed to color all vertices such that no two adjacent vertices are of the same color) can be applied in solving practical tasks as diverse as pattern matching, scheduling jobs to machines, allocating registers in compiler optimization, and even solving Sudoku puzzles. Typically, however, the underlying graphs are subject to (often minor) changes. To make these applications of graph parameters robust, it is important to know which graphs are stable for them in the sense that adding or deleting single edges or vertices does not change them. We initiate the study of stability of graphs for such parameters in terms of their computational complexity. We show that, for various central graph parameters, the problem of determining whether or not a given graph is stable is complete for ???, a well-known complexity class in the second level of the polynomial hierarchy, which is also known as "parallel access to NP.
Filtration-Domination in Bifiltered Graphs
Bifiltered graphs are a versatile tool for modelling relations between data
points across multiple grades of a two-dimensional scale. They are especially
popular in topological data analysis, where the homological properties of the
induced clique complexes are studied. To reduce the large size of these clique
complexes, we identify filtration-dominated edges of the graph, whose removal
preserves the relevant topological properties. We give two algorithms to detect
filtration-dominated edges in a bifiltered graph and analyze their complexity.
These two algorithms work directly on the bifiltered graph, without first
extracting the clique complexes, which are generally much bigger. We present
extensive experimental evaluation which shows that in most cases, more than 90%
of the edges can be removed. In turn, we demonstrate that this often leads to a
substantial speedup, and reduction in the memory usage, of the computational
pipeline of multiparameter topological data analysis
Dominating sets and connected dominating sets in dynamic graphs
In this paper we study the dynamic versions of two basic graph problems: Minimum Dominating Set and its variant Minimum Connected Dominating Set. For those two problems, we present algorithms that maintain a solution under edge insertions and edge deletions in time O( 06\ub7polylog n) per update, where 06 is the maximum vertex degree in the graph. In both cases, we achieve an approximation ratio of O(log n), which is optimal up to a constant factor (under the assumption that P 6= NP). Although those two problems have been widely studied in the static and in the distributed settings, to the best of our knowledge we are the first to present efficient algorithms in the dynamic setting. As a further application of our approach, we also present an algorithm that maintains a Minimal Dominating Set in O(min( 06, m)) per update
Complexity of Stability
Graph parameters such as the clique number, the chromatic number, and the
independence number are central in many areas, ranging from computer networks
to linguistics to computational neuroscience to social networks. In particular,
the chromatic number of a graph (i.e., the smallest number of colors needed to
color all vertices such that no two adjacent vertices are of the same color)
can be applied in solving practical tasks as diverse as pattern matching,
scheduling jobs to machines, allocating registers in compiler optimization, and
even solving Sudoku puzzles. Typically, however, the underlying graphs are
subject to (often minor) changes. To make these applications of graph
parameters robust, it is important to know which graphs are stable for them in
the sense that adding or deleting single edges or vertices does not change
them. We initiate the study of stability of graphs for such parameters in terms
of their computational complexity. We show that, for various central graph
parameters, the problem of determining whether or not a given graph is stable
is complete for \Theta_2^p, a well-known complexity class in the second level
of the polynomial hierarchy, which is also known as "parallel access to NP.
Information Diffusion on Social Networks
In this thesis we model the diffusion of information on social networks. A game
played on a specific type of graph generator, the iterated local transitivity
model, is examined. We study how the dynamics of the game change as the
graph grows, and the relationship between properties of the game on a graph
initially and properties of the game later in the graphâs development. We
show that, given certain conditions, for the iterated local transitivity model
it is possible to predict the existence of a Nash equilibrium at any point in
the graphâs growth. We give sufficient conditions for the existence of Nash
Equilibria on star graphs, cliques and trees. We give some results on potential
games on the iterated local transitivity model.
Chapter 2 provides an introduction to graph properties, and describes various
early graph models. Chapter 3 describes some models for online social
networks, and introduces the iterated local transitivity model which we use
later in the thesis. In Chapter 4 various models for games played on networks
are examined. We study a model for competitive information diffusion on
star graphs, cliques and trees, and we provide conditions for the existence of
Nash Equilibria on these. This model for competitive information diffusion is
studied in detail for the iterated local transitivity model in Chapter 5. We
discuss potential games in Chapter 6 and their existence on the iterated local
transitivity model. We conclude with some suggestions on how to extend and
develop upon the work done in this thesis
Distances and Domination in Graphs
This book presents a compendium of the 10 articles published in the recent Special Issue âDistance and Domination in Graphsâ. The works appearing herein deal with several topics on graph theory that relate to the metric and dominating properties of graphs. The topics of the gathered publications deal with some new open lines of investigations that cover not only graphs, but also digraphs. Different variations in dominating sets or resolving sets are appearing, and a review on some networksâ curvatures is also present
Automated Conjecturing Approach for Benzenoids
Benzenoids are graphs representing the carbon structure of molecules, defined by a closed path in the hexagonal lattice. These compounds are of interest to chemists studying existing and potential carbon structures. The goal of this study is to conjecture and prove relations between graph theoretic properties among benzenoids. First, we generate conjectures on upper bounds for the domination number in benzenoids using invariant-defined functions. This work is an extension of the ideas to be presented in a forthcoming paper. Next, we generate conjectures using property-defined functions. As the title indicates, the conjectures we prove are not thought of on our own, rather generated by a process of automated conjecture-making. This program, named Cá´É´á´á´á´á´á´Ęɪɴɢ, is developed by Craig Larson and Nico Van Cleemput
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