Clique-width for graph classes closed under complementation.

Clique-width is an important graph parameter due to its algorithmic and structural properties. A graph class is hereditary if it can be characterized by a (not necessarily finite) set H of forbidden induced subgraphs. We initiate a systematic study into the boundedness of clique-width of hereditary graph classes closed under complementation. First, we extend the known classification for the |H|=1 case by classifying the boundedness of clique-width for every set H of self-complementary graphs. We then completely settle the |H|=2 case. In particular, we determine one new class of (H1, complement of H1)-free graphs of bounded clique-width (as a side effect, this leaves only six classes of (H1, H2)-free graphs, for which it is not known whether their clique-width is bounded). Once we have obtained the classification of the |H|=2 case, we research the effect of forbidding self-complementary graphs on the boundedness of clique-width. Surprisingly, we show that for a set F of self-complementary graphs on at least five vertices, the classification of the boundedness of clique-width for ({H1, complement of H1} + F)-free graphs coincides with the one for the |H|=2 case if and only if F does not include the bull (the only non-empty self-complementary graphs on fewer than five vertices are P_1 and P_4, and P_4-free graphs have clique-width at most 2). Finally, we discuss the consequences of our results for COLOURING

Randomized Communication and Implicit Representations for Matrices and Graphs of Small Sign-Rank

We prove a characterization of the structural conditions on matrices of sign-rank 3 and unit disk graphs (UDGs) which permit constant-cost public-coin randomized communication protocols. Therefore, under these conditions, these graphs also admit implicit representations. The sign-rank of a matrix $M \in \{\pm 1\}^{N \times N}$ is the smallest rank of a matrix $R$ such that $M_{i,j} = \mathrm{sign}(R_{i,j})$ for all $i,j \in [N]$; equivalently, it is the smallest dimension $d$ in which $M$ can be represented as a point-halfspace incidence matrix with halfspaces through the origin, and it is essentially equivalent to the unbounded-error communication complexity. Matrices of sign-rank 3 can achieve the maximum possible bounded-error randomized communication complexity $\Theta(\log N)$, and meanwhile the existence of implicit representations for graphs of bounded sign-rank (including UDGs, which have sign-rank 4) has been open since at least 2003. We prove that matrices of sign-rank 3, and UDGs, have constant randomized communication complexity if and only if they do not encode arbitrarily large instances of the Greater-Than communication problem, or, equivalently, if they do not contain arbitrarily large half-graphs as semi-induced subgraphs. This also establishes the existence of implicit representations for these graphs under the same conditions.Comment: 28 page

Sliding window temporal graph coloring

Graph coloring is one of the most famous computational problems with applications in a wide range of areas such as planning and scheduling, resource allocation, and pattern matching. So far coloring problems are mostly studied on static graphs, which often stand in contrast to practice where data is inherently dynamic. A temporal graph has an edge set that changes over time. We present a natural temporal extension of the classical graph coloring problem. Given a temporal graph and integers k and Δ, we ask for a coloring sequence with at most k colors for each vertex such that in every time window of Δ consecutive time steps, in which an edge is present, this edge is properly colored at least once. We thoroughly investigate the computational complexity of this temporal coloring problem. More specifically, we prove strong computational hardness results, complemented by efficient exact and approximation algorithms

Succinct Permutation Graphs

We present a succinct data structure for permutation graphs, and their superclass of circular permutation graphs, i.e., data structures using optimal space up to lower order terms. Unlike concurrent work on circle graphs (Acan et al. 2022), our data structure also supports distance and shortest-path queries, as well as adjacency and neighborhood queries, all in optimal time. We present in particular the first succinct exact distance oracle for (circular) permutation graphs. A second succinct data structure also supports degree queries in time independent of the neighborhood's size at the expense of an $O(\log n/\log \log n)$-factor overhead in all running times. Furthermore, we develop a succinct data structure for the class of bipartite permutation graphs. We demonstrate how to run algorithms directly over our succinct representations for several problems on permutation graphs: Clique, Coloring, Independent Set, Hamiltonian Cycle, All-Pair Shortest Paths, and others. Finally, we initiate the study of semi-distributed graph representations; a concept that smoothly interpolates between distributed (labeling schemes) and centralized (standard data structures). We show how to turn some of our data structures into semi-distributed representations by storing only $O(n)$ bits of additional global information, circumventing the lower bound on distance labeling schemes for permutation graphs

PREVENTION OF MEDIATION IN BRIBERY

The relevance. According to statistical data, the observed growth of bribery intermediation in 2022 in relation to 2020 showed the following positive value - 29.63 %. Based on the fact that mediation in bribery in its legal nature resembles the institution of complicity and is a link between other corruption crimes regulated by Articles 290, 291 of the Criminal Code of the Russian Federation, it should be noted that for one fact of criminal mediation there are three cases of taking and giving bribes. We believe that this phenomenon is explained by the high latency of this crime. The very growth of acts under article 291.1 of the Criminal Code of the Russian Federation is due to the lack of countermeasures and, above all, prevention of this crime. The main goal. To determine the specifics and problems of the system of prevention of mediation in bribery. The problems under consideration. The available plans and programs of combating corruption do not provide general, special or individual measures to counteract mediation in bribery, despite the fact that this crime is isolated as an independent one 12 years ago. Scientific analysis of existing normative, planned and methodological documents defining general and specific ways of prevention of mediation in bribery, carried out on the example of acts of the Republic of Khakassia, confirms it. The methods used. In this article we used general scientific methods of knowledge: deduction and analysis, as well as special methods of research: systemic, formal-legal, comparative legal. Conclusions. The author comes to the conclusion that it is necessary to develop a comprehensive system of preventive measures to counteract the commission of crimes regulated by Article 291.1 of the Criminal Code of the Russian Federation only in conjunction with preventive measures for the commission of giving and receiving a bribe, since the legislator, although formally identified mediation in bribery as an independent corpus delicti, but in its illegal meaning it it is of a derivative nature, contributing to the implementation of criminal agreements between the bribe giver and the bribe recipient, but not replacing the

Temporal vertex cover with a sliding time window.

Modern, inherently dynamic systems are usually characterized by a network structure which is subject to discrete changes over time. Given a static underlying graph, a temporal graph can be represented via an assignment of a set of integer time-labels to every edge, indicating the discrete time steps when this edge is active. While most of the recent theoretical research on temporal graphs focused on temporal paths and other “path-related” temporal notions, only few attempts have been made to investigate “non-path” temporal problems. In this paper we introduce and study two natural temporal extensions of the classical problem VERTEX COVER. We present a thorough investigation of the computational complexity and approximability of these two temporal covering problems. We provide strong hardness results, complemented by approximation and exact algorithms. Some of our algorithms are polynomial-time, while others are asymptotically almost optimal under the Exponential Time Hypothesis (ETH) and other plausible complexity assumptions

Adjacency Labeling Schemes for Small Classes

A graph class admits an implicit representation if, for every positive integer n, its n-vertex graphs have a O(log n)-bit (adjacency) labeling scheme, i.e., their vertices can be labeled by binary strings of length O(log n) such that the presence of an edge between any pair of vertices can be deduced solely from their labels. The famous Implicit Graph Conjecture posited that every hereditary (i.e., closed under taking induced subgraphs) factorial (i.e., containing 2^O(n log n) n-vertex graphs) class admits an implicit representation. The conjecture was recently refuted [Hatami and Hatami, FOCS '22], and does not even hold among monotone (i.e., closed under taking subgraphs) factorial classes [Bonnet et al., ICALP '24]. However, monotone small (i.e., containing at most n! cⁿ many n-vertex graphs for some constant c) classes do admit implicit representations. This motivates the Small Implicit Graph Conjecture: Every hereditary small class admits an O(log n)-bit labeling scheme. We provide evidence supporting the Small Implicit Graph Conjecture. First, we show that every small weakly sparse (i.e., excluding some fixed bipartite complete graph as a subgraph) class has an implicit representation. This is a consequence of the following fact of independent interest proved in the paper: Every weakly sparse small class has bounded expansion (hence, in particular, bounded degeneracy). Second, we show that every hereditary small class admits an O(log³ n)-bit labeling scheme, which provides a substantial improvement of the best-known polynomial upper bound of n^(1-ε) on the size of adjacency labeling schemes for such classes. This is a consequence of another fact of independent interest proved in the paper: Every small class has neighborhood complexity O(n log n)