### Working paper

## A complexity dichotomy for the dominating set problem

This book constitutes the proceedings of the 13th International Computer Science Symposium in Russia, CSR 2018, held in Moscow, Russia, in May 2018.

The 24 full papers presented together with 7 invited lectures were carefully reviewed and selected from 42 submissions. The papers cover a wide range of topics such as algorithms and data structures; combinatorial optimization; constraint solving; computational complexity; cryptography; combinatorics in computer science; formal languages and automata; algorithms for concurrent and distributed systems; networks; and proof theory and applications of logic to computer science.

A graph is König for a q-path if every its induced subgraph has the following property. The maximum number of pairwise vertex-disjoint induced paths each on q vertices is equal to the minimum number of vertices, such that removing all the vertices produces a graph having no an induced path on q vertices. In this paper, for every q>4, we describe all Konig graphs for a q-path obtained from forests and simple sycles by replacing some vertices into graphs not containing induced paths on q vertices.

For a graph property X, let X_{n} be the number of graphs with vertex set {1, . . . , n} having property X, also known as the speed of X. A property X is called factorial if X is hereditary (i.e. closed under taking induced subgraphs) and n^{c1n} ≤ X_{n} ≤ n^{c2n} for some constants c_{1} and c_{2}. Hereditary properties with speed slower than factorial are surprisingly well structured. The situation with factorial properties is more complicated and less explored. Only the properties with speeds up to the Bell number are well studied and well behaved. To better understand the behavior of factorial properties with faster speeds we introduce a structural tool called locally bounded coverings and show that a variety of graph properties can be described by means of this tool.

For a graph property X, let X_{n} be the set of graphs with the vertex set {1, . . . , n} that satisfy the property X. A property X is called factorial if X is hereditary (i. e. closed under taking induced subgraphs) and n^{c1n} ≤ X ≤ n^{c2n} for some positive constants c_{1} and c_{2}. A graph G is a *quasi-line* if for every vertex v, the set of neighbors of v can be expressed as a union of two cliques. In the present paper we identify almost all factorial subclasses of quasi-line graphs defined by one forbidden induced subgraph. We use these new results to prove that the class Free(K_{1,3},W_{4}) is factorial, which improves on a result of Lozin, Mayhill and Zamaraev [8].

We investigate regular realizability (RR) problems, which are the prob- lems of verifying whether intersection of a regular language – the input of the problem – and fixed language called filter is non-empty. In this pa- per we focus on the case of context-free filters. Algorithmic complexity of the RR problem is a very coarse measure of context-free languages com- plexity. This characteristic is compatible with rational dominance. We present examples of P-complete RR problems as well as examples of RR problems in the class NL. Also we discuss RR problems with context- free filters that might have intermediate complexity. Possible candidates are the languages with polynomially bounded rational indices.

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.

We study the following computational problem: for which values of k, the majority of n bits MAJn can be computed with a depth two formula whose each gate computes a majority function of at most k bits? The corresponding computational model is denoted by MAJk o MAJk. We observe that the minimum value of k for which there exists a MAJk o MAJk circuit that has high correlation with the majority of n bits is equal to Θ(n1/2). We then show that for a randomized MAJk o MAJk circuit computing the majority of n input bits with high probability for every input, the minimum value of k is equal to n2/3+o(1). We show a worst case lower bound: if a MAJk o MAJk circuit computes the majority of n bits correctly on all inputs, then k ≥ n13/19+o(1). This lower bound exceeds the optimal value for randomized circuits and thus is unreachable for pure randomized techniques. For depth 3 circuits we show that a circuit with k = O(n2/3) can compute MAJn correctly on all inputs.

This book constitutes the refereed proceedings of the 44th International Conference on Current Trends in Theory and Practice of Computer Science, SOFSEM 2018, held in Krems, Austria, in January/February 2018. The 48 papers presented in this volume were carefully reviewed and selected from 97 submissions. They were organized in topical sections named: foundations of computer science; software engineering: advances methods, applications, and tools; data, information and knowledge engineering; network science and parameterized complexity; model-based software engineering; computational models and complexity; software quality assurance and transformation; graph structure and computation; business processes, protocols, and mobile networks; mobile robots and server systems; automata, complexity, completeness; recognition and generation; optimization, probabilistic analysis, and sorting; filters, configurations, and picture encoding; machine learning; text searching algorithms; and data model engineering.

For a graph property X, let X_{n} be the number of graphs with vertex set {1, . . . , n} having property X, also known as the speed of X. A property X is called factorial if X is hereditary (i.e., closed under taking induced subgraphs) and n^{c1n} ≤ X_{n} ≤ n^{c2n} for some positive constants c_{1} and c_{2}. Hereditary properties with speed slower than factorial are surprisingly well structured. The situation with factorial properties is more complicated and less explored. To better understand the structure of factorial properties we look for minimal superfactorial ones. In [J.P. Spinrad, Nonredundant 1’s in *Γ*-free matrices, SIAM J. Discrete Math. 8 (1995) 251–257], Spinrad showed that the number of n-vertex chordal bipartite graphs is 2^{Θ(n log2n)}, which means that this class is superfactorial. On the other hand, all subclasses of chordal bipartite graphs that have been studied in the literature, such as forest, bipartite permutation, bipartite distance-hereditary or convex graphs, are factorial. In this paper, we study more hereditary subclasses of chordal bipartite graphs and reveal both factorial and superfactorial members in this family. The latter fact shows that the class of chordal bipartite graphs is not a minimal superfactorial one. Finding minimal superfactorial classes in this family remains a challenging open question.