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## On the complexity of failed zero forcing

Let *G* be a simple graph whose vertices are partitioned into two subsets, called ‘filled’ vertices and ‘empty’ vertices. A vertex *v* is said to be forced by a filled vertex *u* if *v* is a unique empty neighbor of *u*. If we can fill all the vertices of *G* by repeatedly filling the forced ones, then we call an initial set of filled vertices a forcing set. We discuss the so-called failed forcing number of a graph, which is the largest cardinality of a set which is not forcing. Answering the recent question of Ansill, Jacob, Penzellna, Saavedra, we prove that this quantity is NP-hard to compute. Our proof also works for a related graph invariant which is called the skew failed forcing number.

The notion of a boundary class is a useful notion in the investigation of the complexity of extremal problems on graphs. One boundary class is known for the independent set problem and three boundary classes are known for the dominating set problem. In this paper it is proved that the set of boundary classes for the 3-colouring problem is infinite.

The notion of a boundary graph property was recently introduced as a relaxation of that of a minimal property and was applied to several problems of both algorithmic and combinatorial nature. In the present paper, we first survey recent results related to this notion and then apply it to two algorithmic graph problems: Hamiltonian cycle and Vertex k-colorability. In particular, we discover the first two boundary classes for the Hamiltonian cycle problem and prove that for any k > 3 there is a continuum of boundary classes for Vertex k-colorability.

Proceedings include extended abstracts of reports presented at the III International Conference on Optimization Methods and Applications “Optimization and application” (OPTIMA-2012) held in Costa da Caparica, Portugal, September 23—30, 2012.

The problem of quick detection of central nodes in large networks is studied. There are many measures that allow to evaluate a topological importance of nodes of the network. Unfortunately, most of them cannot be applied to large networks due to their high computational complexity. However, if we narrow the initial network and apply these centrality measures to the sparse network, it is possible that the obtained set of central nodes will be similar to the set of central nodes in large networks. If these sets are similar, the centrality measures with a high computational complexity can be used for central nodes detection in large networks. To check the idea, several random networks were generated and different techniques of network reduction were considered. We also adapted some rules from social choice theory for the key nodes detection. As a result, we show how the initial network should be narrowed in order to apply centrality measures with a high computational complexity and maintain the set of key nodes of a large network.

The collection represents proceedings of the nineth international conference "Discrete Models in Control Systems Theory" that is held by Lomonosov Moscow State Uneversity and is dedicated in 90th anniversary of Sergey Vsevolodovich Yablonsky's birth. The conference subject are includes: discrete functional systems; discrete functions properties; control systems synthesis, complexity, reliability, and diagnostics; automata; graph theory; combinatorics; coding theory; mathematical methods of information security; theory of pattern recognition; mathematical theory of intellegence systems; applied mathematical logic. The conference is sponsored by Russian Foundation for Basic Research (project N 15-01-20193-г).

This book constitutes the refereed proceedings of the 23rd Annual Symposium on Combinatorial Pattern Matching, CPM 2012, held in Helsinki, Finalnd, in July 2012. The 33 revised full papers presented together with 2 invited talks were carefully reviewed and selected from 60 submissions. The papers address issues of searching and matching strings and more complicated patterns such as trees, regular expressions, graphs, point sets, and arrays. The goal is to derive non-trivial combinatorial properties of such structures and to exploit these properties in order to either achieve superior performance for the corresponding computational problems or pinpoint conditions under which searches cannot be performed efficiently. The meeting also deals with problems in computational biology, data compression and data mining, coding, information retrieval, natural language processing, and pattern recognition.

The notion of a boundary class is a useful notion in the investigation of the complexity of extremal problems on graphs. One boundary class is known for the independent set problem and three boundary classes are known for the dominating set problem. In this paper it is proved that the set of boundary classes for the 3-colouring problem is infinite.

*When a society needs to take a collective decision one could apply some aggregation method, particularly, voting. One of the main problems with voting is manipulation. We say a voting rule is vulnerable to manipulation if there exists at least one voter who can achieve a better voting result by misrepresenting his or her preferences. The popular approach to comparing manipulability of voting rules is defining complexity class of the corresponding manipulation problem. This paper provides a survey into manipulation complexity literature considering variety of problems with different assumptions and restrictions.*

This proceedings publication is a compilation of selected contributions from the “Third International Conference on the Dynamics of Information Systems” which took place at the University of Florida, Gainesville, February 16–18, 2011. The purpose of this conference was to bring together scientists and engineers from industry, government, and academia in order to exchange new discoveries and results in a broad range of topics relevant to the theory and practice of dynamics of information systems. Dynamics of Information Systems: Mathematical Foundation presents state-of-the art research and is intended for graduate students and researchers interested in some of the most recent discoveries in information theory and dynamical systems. Scientists in other disciplines may also benefit from the applications of new developments to their own area of study.

A form for an unbiased estimate of the coefficient of determination of a linear regression model is obtained. It is calculated by using a sample from a multivariate normal distribution. This estimate is proposed as an alternative criterion for a choice of regression factors.