Electronic Notes in Discrete Mathematics最新文献

In this work we compute the group inverse of the Laplacian of the connections of two networks by and edge in terms of the Laplacians of the original networks. Thus the effective resistances and Kirchhoff index of the new network can be derived from the Kirchhoff indexes of the original networks.

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.

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 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 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 $n\to \infty$.

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 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.

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.

We consider decision models associated with cooperative influence games, the oblivious and the non-oblivious influence models. In those models the satisfaction and the power measures were introduced and studied. We analyze the computational complexity of those measures when the influence level is set to unanimity and the rule of decision is simple majority. We show that computing the satisfaction and the power measure in those systems are #P-hard.