On a lower bound for the number of vertices of an integral graph with a given diameter

Context. This article describes how to solve the game problem of assigning staff to work on projects based on an ontological approach. The essence of the problem is this. There is a need to create teams to carry out several projects. Each project is defined by a set of necessary ontological knowledge. To implement projects, managers invite qualified specialists (agents), whose abilities are also defined by sets of ontologies. The composition of the teams should be such that the combined ontologies of their agents cover the set of ontologies of the respective projects. Each agent with a certain probability can take part in the implementation of several projects. Simultaneous work of the agent on different projects is not allowed. It is necessary to determine the order of project implementation and the corresponding order of personnel appointment.
 Objective of the study is to develop a mathematical model of stochastic game, recurrent Markov methods for its solution, algorithmic and software, computer experiment, analysis of results and development of recommendations for their practical application.
 Method. A stochastic game algorithm for coloring an undirected random graph was used to plan project execution. To do this, the number of vertices of the graph is taken equal to the number of projects. The edges of the project graph for which the same agent is invited are connected by edges. Due to the recovery failures of agents, the connections between the vertices of the graph change dynamically. It is necessary to achieve the correct coloring of the random graph. Then projects with the same colored vertices of the graph can be executed in parallel, and projects with different colors of vertices – in series.
 Results. The article builds a mathematical model of a stochastic game and a self-learning Markov method for its solution. Each vertex of the graph is controlled by the player. The player’s pure strategies are the elements of the color palette. After selecting the color of their own top, each player calculates the current loss as a relative number of identical colors in the local set of neighboring players. The goal of the players is to minimize the functions of average losses. The Markov recurrent method provides an adaptive choice of colors for the vertices of a random graph based on dynamic vectors of mixed strategies, the values of which depend on the current losses of players. The result of a stochastic game is an asymptotically correctly colored random graph, when each edge of the initial deterministic graph will correspond on average to different colors of vertices.
 Conclusions. A computer experiment was performed, which confirmed the convergence of the stochastic game for the problem of coloring a random graph. This made it possible to determine the procedure for appointing staff to implement projects.

In this paper we consider the Steiner problem in graphs which is the problem of connecting together, at minimum cost, a number of vertices in an undirected graphs. We present a formulation of the problem as a shortest spanning tree (SST) problem with additional constraints. By relaxing thses additional constraints in a lagrangean fashion we obtain a lower bound for the problem based upon the solution of an unconstrained SST problem. Problem reduction tests derived from both the original problem and the lagrangean relaxation are given. Incorporating the bound and the reduction tests into a tree search procedure enables us to solve problems involving the connection of up to 1250 vertices in a graph with 62500 edges and 2500 vertices.

In this paper we consider the Steiner problem in graphs. This is the problem of connecting together, at minimum cost, a number of vertices in an undirected graph. We present two lower bounds for the problem, these bounds being based on two separate Lagrangian relaxations of a zero‐one integer programming formulation of the problem. Problem reduction tests derived from both the original problem and the Lagrangian relaxations are given. Incorporating the bounds and the reduction tests into a tree search procedure enables us to solve problems involving the connection of up to 50 vertices in a graph with 200 undirected arcs and 100 vertices.

We propose a Ramsey approach to the dimensional analysis of physical systems, which complements the seminal Buckingham theorem. Dimensionless constants describing a given physical system are represented as vertices of a graph, referred to as a dimensions graph. Two vertices are connected by an aqua-colored edge if they share at least one common dimensional physical quantity and by a brown edge if they do not. In this way, a bi-colored complete Ramsey graph is obtained. The relations introduced between the vertices of the dimensions graph are non-transitive. According to the Ramsey theorem, a monochromatic triangle must necessarily appear in a dimensions graph constructed from six vertices, regardless of the order of the vertices. Mantel–Turán analysis is applied to study these graphs. The proposed Ramsey approach is extended to graphs constructed from fundamental physical constants. A physical interpretation of the Ramsey analysis of dimensions graphs is suggested. A generalization of the proposed Ramsey scheme to multi-colored Ramsey graphs is also discussed, along with an extension to infinite sets of dimensionless constants. The introduced dimensions graphs are invariant under rotations of reference frames, but they are sensitive to Galilean and Lorentz transformations. The coloring of the dimensions graph is independent of the chosen system of units. The number of vertices in a dimensions graph is relativistically invariant and independent of the system of units.

Efficiently processing queries against very large graphs is an important research topic largely driven by emerging real world applications, as diverse as XML databases, GIS, web mining, social network analysis, ontologies, and bioinformatics. In particular, graph reachability has attracted a lot of research attention as reachability queries are not only common on graph databases, but they also serve as fundamental operations for many other graph queries. The main idea behind answering reachability queries in graphs is to build indices based on reachability labels. Essentially, each vertex in the graph is assigned with certain labels such that the reachability between any two vertices can be determined by their labels. Several approaches have been proposed for building these reachability labels; among them are interval labeling (tree cover) and 2-hop labeling. However, due to the large number of vertices in many real world graphs (some graphs can easily contain millions of vertices), the computational cost and (index) size of the labels using existing methods would prove too expensive to be practical. In this paper, we introduce a novel graph structure, referred to as path-tree, to help labeling very large graphs. The path-tree cover is a spanning subgraph of G in a tree shape. We demonstrate both analytically and empirically the effectiveness of our new approaches.

An efficient certifying algorithm for the Hamiltonian cycle problem on circular-arc graphs

The lattice cohomology of a plumbed 3-manifold M associated with a connected negative definite plumbing graph is an important tool in the study of topological properties of M and in the comparison of the topological properties with analytic ones, whenever M is realized as complex analytic singularity link. By definition, its computation is based on the (Riemann–Roch) weights of the lattice points of |${\mathbb Z}^s$|⁠, where s is the number of vertices of the plumbing graph. The present article reduces the rank of this lattice to the number of “bad” vertices of the graph. Usually, the geometry/topology of M is codified exactly by these “bad” vertices via surgery or other constructions. Their number measures how far is the plumbing graph from a rational one (or, how far is M from an L-space). The effect of the reduction appears also at the level of certain multivariable (topological Poincaré) series. Since from these series, one can also read the Seiberg–Witten invariants, the Reduction Theorem provides new formulae for these invariants too. The reduction also implies the vanishing |${\mathbb H}^q=0$| of the lattice cohomology for q≥ν, where ν is the number of “bad” vertices. (This bound is sharp.)

Graph matching, as one of the most fundamental problems in graph database, has a wide range of applications. Due to the large scale of graph database and the hardness of graph matching, it is an attractive alternative to make use of the cloud to store massive data graphs and conduct the complex graph matching. To protect the privacy, data graphs are usually encrypted before being outsourced to the cloud. A few schemes have been proposed to support graph matching query over encrypted graphs. However, none of them can realize efficient subgraph extraction when the matched subgraph needs to be exactly located at the data graph. The graph user has to perform the complex subgraph isomorphism operation to extract the isomorphic subgraph from the matched data graph in state-of-the-art schemes. The time complexity of judging subgraph isomorphism is <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$O(n!n^{2})$</tex-math></inline-formula> at most, where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$n$</tex-math></inline-formula> is the number of vertices in the data graph. In order to solve this problem, we propose a privacy-preserving graph matching query scheme supporting quick subgraph extraction in this paper. In our design, two non-colluding cloud servers are adopted to accomplish the matching operation jointly. Neither of them can infer the plaintexts of graphs. The first cloud server can prune some impossible matched vertices from the data graph, and get a possible matched matrix to represent which vertices in the data graph might match with the vertices in the query graph. The second cloud server performs the related matrix operations based on its stored information and the information sent by the first cloud server. The two cloud servers jointly verify the correctness of the matched matrix. The graph user can directly and quickly extract the matrix of the subgraph isomorphic to the query graph from the data graph matrix based on the matched matrix, and further recover this subgraph. No subgraph isomorphism operation is involved in the procedure of subgraph extraction for the graph user. The time complexity of subgraph extraction is <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$O(m^{2})$</tex-math></inline-formula> in our scheme, where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$m$</tex-math></inline-formula> is the number of vertices in the query graph. The extensive experiments with real-world database demonstrate the efficiency of the proposed privacy-preserving graph matching scheme.

The maximum cut (MAX-CUT) problem is to find a bipartition of the vertices in a given graph such that the number of edges with ends in different sets reaches the largest. Though, several experimental investigations have shown that evolutionary algorithms (EAs) are efficient for this NP-complete problem, there is little theoretical work about EAs on the problem. In this paper, we theoretically investigate the performance of EAs on the MAX-CUT problem. We find that both the (1+1) EA and the (1+1) EA*, two simple EAs, efficiently achieve approximation solutions of (m/2)+(1/4)s(G) and (m/2)+(1/2)(√{8m+1}-1), where m and s(G) are respectively the number of edges and the number of odd degree vertices in the input graph. We also reveal that for a given integer k the (1+1) EA* finds a cut of size at least k in expected runtime O(nm+1/δ(4k)) and a cut of size at least (m/2)+k in expected runtime O(n(2)m+1/δ((64/3)k(2))), where δ is a constant mutation probability and n is the number of vertices in the input graph. Finally, we show that the (1+1) EA and the (1+1) EA* are better than some local search algorithms in one instance, and we also show that these two simple EAs may not be efficient in another instance.

ABSTRACTA well‐known theorem of Mantel states that every ‐vertex graph with more than edges contains a triangle. An interesting problem in extremal graph theory studies the minimum number of edges contained in triangles among graphs with a prescribed number of vertices and edges. Erdős, Faudree, and Rousseau (1992) showed that a graph on vertices with more than edges contains at least edges in triangles. Such edges are called triangular edges. In this paper, we present a spectral version of the result of Erdős, Faudree, and Rousseau. Using the supersaturation‐stability and the spectral technique, we prove that every ‐vertex graph with contains at least triangular edges, unless is a balanced complete bipartite graph. The method in our paper has some interesting applications. Firstly, the supersaturation‐stability can be used to revisit a conjecture of Erdős concerning the booksize of a graph, which was initially proved by Edwards (unpublished), and independently by Khadžiivanov and Nikiforov (1979). Secondly, our method can improve the bound on the order of the spectral extremal graph when we forbid the friendship graph as a substructure. We drop the condition that requires the order to be sufficiently large, which was investigated by Cioabă et al. (2020) using the triangle removal lemma. Thirdly, this method can be utilized to deduce the classical stability for odd cycles, and it gives more concise bounds on parameters. Finally, supersaturation stability could be applied to deal with the spectral graph problems on counting triangles, which was recently studied by Ning and Zhai (2023).

A branch vertex in a tree is a vertex of degree at least three. We study the NP-hard problem of constructing spanning trees with as few branch vertices as possible. This problem generalizes the famous Hamiltonian Path problem which corresponds to the case of no vertices having degree three or more. It has been extensively studied in the literature and has important applications in network design and optimization. In this paper, we study the problem of finding a spanning tree with the minimum number of branch vertices in graphs of bounded neighborhood diversity. Neighborhood diversity, a generalization of vertex cover to dense graphs, plays an important role in the design of algorithms for such graphs.

The concepts of nth degrees and nth-order odd vertices in graphs are introduced. The first degree of a vertex v in a graph G is the degree of v, while the nth degree ($n\geqq 2$) of $v $ is the sum of the $(n - 1)$st degrees of the vertices adjacent to $v $ in G. By a first-order odd vertex in a graph G is meant an (ordinary) odd vertex in G, while for $n\geqq 2$, an nth-order odd vertex of G is a vertex adjacent to an odd number of $(n - 1)$st-order odd vertices. The number of nth-order odd vertices, $n = 1,2, \cdots $, is investigated. A sequence $s_{1}, s_{2}, \cdots ,s_n , \cdots $ of integers is defined to be a generalized odd vertex sequence if there exists a graph G containing exactly $s_{n}$nth-order odd vertices for every positive integer n. Generalized odd vertex sequences are characterized. Relationships between the nth degrees of the vertices of a graph G and the walks of length n in G are described. The analogous problem for digraphs is also discussed.

This paper studies the Maximum Internal Spanning Tree problem which is to find a spanning tree with the maximum number of internal vertices on a graph. We prove that the problem can be solved in polynomial time on interval graphs. The idea is based on the observation that the number of internal vertices in a maximum internal spanning tree is at most one less than the number of edges in a maximum path cover on any graph. On an interval graph, we present an \(O(n^{2})\)-algorithm to find a spanning tree in which the number of internal vertices is exactly one less than the number of edges in a maximum path cover of the graph, where n is the number of vertices in the interval graph.

A straight-line drawing $\delta$ of a planar graph $G$ need not be plane, but can be made so by \emph{untangling} it, that is, by moving some of the vertices of $G$. Let shift$(G,\delta)$ denote the minimum number of vertices that need to be moved to untangle $\delta$. We show that shift$(G,\delta)$ is NP-hard to compute and to approximate. Our hardness results extend to a version of \textsc{1BendPointSetEmbeddability}, a well-known graph-drawing problem. Further we define fix$(G,\delta)=n-shift(G,\delta)$ to be the maximum number of vertices of a planar $n$-vertex graph $G$ that can be fixed when untangling $\delta$. We give an algorithm that fixes at least $\sqrt{((\log n)-1)/\log \log n}$ vertices when untangling a drawing of an $n$-vertex graph $G$. If $G$ is outerplanar, the same algorithm fixes at least $\sqrt{n/2}$ vertices. On the other hand we construct, for arbitrarily large $n$, an $n$-vertex planar graph $G$ and a drawing $\delta_G$ of $G$ with fix$(G,\delta_G) \le \sqrt{n-2}+1$ and an $n$-vertex outerplanar graph $H$ and a drawing $\delta_H$ of $H$ with fix$(H,\delta_H) \le 2 \sqrt{n-1}+1$. Thus our algorithm is asymptotically worst-case optimal for outerplanar graphs.

Solving the maximum internal spanning tree problem on interval graphs in polynomial time