Electronic Notes in Discrete Mathematics最新文献

In this paper we describe the automorphism groups of graphs and edge-colored graphs that are cyclic as permutation groups. In addition, we show that every such group is the automorphism group of a complete graph whose edges are colored with 3 colors, and we characterize those groups that are automorphism groups of simple graphs.

In this work we consider the shortest path problem and the single facility Weber location problem in any real space of finite dimension where there exist different types of polyhedral obstacles or forbidden regions. These regions are polyhedral sets and the metric considered in the space is the Manhattan metric. We present a result that reduce these continuous problems into problems in a “add hoc” graph, where the original problems can be solved using elementary techniques of Graph Theory. We show that, fixed the dimension of the space, both the reduction and the resolution can be done in polynomial time.

We introduce a new family of hyperplane arrangements in dimension $n\ge 3$ that includes both the Shi arrangement and the Ish arrangement. We prove that all the members of this family have the same number of regions — the connected components of the complement of the union of the hyperplanes — which can be bijectively labeled with the Pak-Stanley labelling. In addition, we characterise the Pak-Stanley labels of the regions of this family of hyperplane arrangements.

We present novel production matrices for non-crossing partitions, connected geometric graphs, and k-angulations, which provide another way of counting the number of such objects. For instance, a formula for the number of connected geometric graphs with given root degree, drawn on a set of n points in convex position in the plane, is presented. Further, we find the characteristic polynomials and we provide a characterization of the eigenvectors of the production matrices.

We deal with a geometric quadrangulation of a polygon P. We define a new notion called “the spirality” of P, which measures how close P is from being a convex polygon. Using the spirality, we describe (1) a condition of P to admit a geometric quadrangulation, and (2) a condition of P guaranteeing that any two geometric quadrangulations on P can be related by a sequence of edge flips.

A free-form Sudoku puzzle is a square arrangement of $m\times m$ cells such that the cells are partitioned into m subsets (called blocks) of equal cardinality. The goal of the puzzle is to place integers $1,\dots m$ in the cells such that the numbers in every row, column and block are distinct. Represent each cell by a vertex and add edges between two vertices exactly when the corresponding cells, according to the rules, must contain different numbers. This yields the associated free-form Sudoku graph. It was shown that all Sudoku graphs are integral graphs, in this paper we present many free-form Sudoku graphs that are still integral graphs.

We consider first order expressible properties of random perfect graphs. That is, we pick a graph $G_n$ uniformly at random from all (labelled) perfect graphs on n vertices and consider the probability that it satisfies some graph property that can be expressed in the first order language of graphs. We show that there exists such a first order expressible property for which the probability that $G_n$ satisfies it does not converge as $n\to \infty$.

We explain how to compute all the solutions of a nonlinear integer problem using the algebraic test-sets associated to some linear subproblem. These test-sets are obtained using Gröbner bases. We compare our method with previous approaches.

In this paper we study the problem of finding a maximum colorful independent set in vertex-colored graphs. Specifically, given a graph with colored vertices, we wish to find an independent set containing the maximum number of colors. Here we aim to give a dichotomy overview on the complexity of this problem. We first show that the problem is NP-hard even for some cases where the maximum independent set problem is easy, such as cographs and P₅-free graphs. Next, we provide polynomial-time algorithms for cluster graphs and trees.

Neural networks present big popularity and success in many fields. The large training time process problem is a very important task nowadays. In this paper, a new approach to get over this issue based on reducing dataset size is proposed. Two algorithms covering two different shape notions are shown and experimental results are given.