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A Comprehensive Grobner system for a parametric ideal I in K(A)[X] represents the collection of all Grobner bases of the ideals I' in K[X] obtained as the values of the parameters A vary in K. The recent algorithms for computing them comprehensive Grobner systems consider the corresponding ideal J in K[A,X], and are based on stability of Grobner bases of ideals under specializations of the parameters. Starting from a Grobner basis of J, the computation splits recursively depending on the vanishing of the evaluation of some ``coefficients'' in K[A]. In this paper, taking inspiration from the algorithm described by Nabeshima, we create a new iterative algorithm to compute comprehensive Grobner systems. We show how we keep track of the sub-cases to be considered, and how we avoid some redundant computation branches using ``comparatively-cheap'' ideal-membership tests, instead of radical-membership tests.

Athanasiadis studied arrangements obtained by adding shifted hyperplanes to the braid arrangement. Similarly, Bailey studied arrangements obtained by adding tilted hyperplanes to the braid arrangement. These two kinds of arrangements are associated with directed graphs and their freeness was characterized in terms of the associated graphs. In addition, there is coincidence of freeness. Namely, if Athanasiadis' arrangement is free, then the corresponding Bailey's arrangement is free, and vice versa. In this paper, we generalize this phenomenon by using gain graphs., Comment: 21 pages

In this article, we study the weak and strong Lefschetz of higher dimensional quotients and dimension 1 almost complete intersections. We then apply the obtained results to the study of the Jacobian algebra of hyperplane arrangements., Comment: arXiv admin note: text overlap with arXiv:1911.04083, arXiv:2003.06294

In this paper, we study the total outer-connected domination number of the middle graph of a simple graph and we obtain tight bounds for this number in terms of the order of the middle graph. We also compute the total outer-connected domination number of some families of graphs, explicitly. Moreover, some Nordhaus-Gaddum-like relations are presented for the total outer-connected domination number of middle graphs.

In this paper, we study the notions of connected domination number and of outer-connected domination number for middle graphs. Indeed, we obtain tight bounds for this number in terms of the order of the graph M(G). We also compute the outer-connected domination number of some families of graphs such as star graphs, cycle graphs, wheel graphs, complete graphs, complete bipartite graphs and some operation on graphs, explicitly. Moreover, some Nordhaus-Gaddum-like relations are presented for the outer-connected domination number of middle graphs.

A total dominating set of a graph G with no isolated vertices is a subset S of the vertex set such that every vertex of G is adjacent to a vertex in S. The total domination number of G is the minimum cardinality of a total dominating set of G. In this paper, we study the total domination number of middle graphs. Indeed, we obtain tight bounds for this number in terms of the order of the graph G. We also compute the total domination number of the middle graph of some known families of graphs explicitly. Moreover, some Nordhaus-Gaddum-like relations are presented for the total domination number of middle graphs., Comment: To appear in the Electronic Journal of Graph Theory and Applications. arXiv admin note: substantial text overlap with arXiv:2008.02975

A total dominator coloring of a graph G is a proper coloring of G in which each vertex of the graph is adjacent to every vertex of some color class. The total dominator chromatic number of a graph is the minimum number of color classes in a total dominator coloring. In this article, we study the total dominator coloring on middle graphs by giving several bounds for the case of general graphs and trees. Moreover, we calculate explicitely the total dominator chromatic number of the middle graph of several known families of graphs.

It is an open question to give a combinatorial interpretation of the Falk invariant of a hyperplane arrangement, i.e. the third rank of successive quotients in the lower central series of the fundamental group of the arrangement. In this article, we give a combinatorial formula for this invariant in the case of hyperplane arrangements that are complete lift representation of certain gain graphs. As a corollary, we compute the Falk invariant for the cone of the braid, Shi, Linial and semiorder arrangements., Comment: To appear in Discrete & Computational Geometry. arXiv admin note: substantial text overlap with arXiv:1707.08449, arXiv:1703.09402

In this paper, we study the domination number of middle graphs. Indeed, we obtain tight bounds for this number in terms of the order of the graph. We also compute the domination number of some families of graphs such as star graphs, double start graphs, path graphs, cycle graphs, wheel graphs, complete graphs, complete bipartite graphs and friendship graphs, explicitly. Moreover, some Nordhaus-Gaddum-like relations are presented for the domination number of middle graphs.

In this article, we study the $k$-Lefschetz properties for non-Artinian algebras, proving that several known results in the Artinian case can be generalized in this setting. Moreover, we describe how to characterize the graded algebras having the $k$-Lefschetz properties using sectional matrices. We then apply the obtained results to the study of the Jacobian algebra of hyperplane arrangements, with particular attention to the class of free arrangements., Comment: arXiv admin note: substantial text overlap with arXiv:1911.04083

In this paper, we study the class of free multiarrangements of hyperplanes. Specifically, we investigate the relations between freeness over a field of finite characteristic and freeness over the rationals., Comment: arXiv admin note: text overlap with arXiv:1803.09908

In combinatorics, a latin square is a $n\times n$ matrix filled with n different symbols, each occurring exactly once in each row and exactly once in each column. Associated to each latin square, we can define a simple graph called a latin square graph. In this article, we compute lower and upper bounds for the domination number and the k-tuple total domination numbers of such graphs. Moreover, we describe a formula for the 2-tuple total domination number., Comment: To appear in Graphs and Combinatorics

In this article, we study the weak and strong Lefschetz properties, and the related notion of almost revlex ideal, in the non-Artinian case, proving that several results known in the Artinian case hold also in this more general setting. We then apply the obtained results to the study of the Jacobian algebra of hyperplane arrangements., Comment: To appear in the Journal of Algebra

We study the classes of free and plus-one generated hyperplane arrangements. Specifically, we describe how to compute the associated prime ideals of the Jacobian ideal of such an arrangement from its lattice of intersection. Moreover, we prove that the localization of a plus-one generated arrangement is free or plus-one generated., Comment: To appear in "Communications in Algebra"

We study the combinatorics of hyperplane arrangements over arbitrary fields. Specifically, we determine in which situation an arrangement and its reduction modulo a prime number have isomorphic lattices via the use of minimal strong $\sigma$-Gr\"obner bases. Moreover, we prove that the Terao's conjecture over finite fields implies the conjecture over the rationals., Comment: Accepted to Advances of Applied Mathematics

A $k$-tuple total dominating set ($k$TDS) of a graph $G$ is a set $S$ of vertices in which every vertex in $G$ is adjacent to at least $k$ vertices in $S$. The minimum size of a $k$TDS is called the $k$-tuple total dominating number and it is denoted by $\gamma_{\times k,t}(G)$. We give a constructive proof of a general formula for $\gamma_{\times 3, t}(K_n \Box K_m)$.

Every subarrangement of Weyl arrangements of type $ B_{\ell} $ is represented by a signed graph. Edelman and Reiner characterized freeness of subarrangements between type $ A_{\ell-1} $ and type $ B_{\ell} $ in terms of graphs. Recently, Suyama and the authors characterized freeness for subarrangements containing Boolean arrangements satisfying a certain condition. This article is a sequel to the previous work. Namely, we give a complete characterization for freeness of arrangements between Boolean arrangements and Weyl arrangements of type $ B_{\ell} $ in terms of graphs., Comment: 15 pages

We introduce the package \textbf{arrangements} for the software CoCoA. This package provides a data structure and the necessary methods for working with hyperplane arrangements. In particular, the package implements methods to enumerate many commonly studied classes of arrangements, perform operations on them, and calculate various invariants associated to them.

In this paper, we study the class of free hyperplane arrangements. Specifically, we investigate the relations between freeness over a field of finite characteristic and freeness over $\mathbb{Q}$.

In this article we describe two new characterizations of freeness for hyperplane arrangements via the study of the generic initial ideal and of the sectional matrix of the Jacobian ideal of arrangements.

The fundamental group of the complement of a hyperplane arrangement in a complex vector space is an important topological invariant. The third rank of successive quotients in the lower central series of the fundamental group was called Falk invariant of the arrangement since Falk gave the first formula and asked to give a combinatorial interpretation. In this article, we give a combinatorial formula for the Falk invariant of hyperplane arrangements attached to certain gain graphs., Comment: To appear in the Australasian Journal of Combinatorics. arXiv admin note: text overlap with arXiv:1703.09402

A Weyl arrangement is the hyperplane arrangement defined by a root system. Arnold and Saito proved that every Weyl arrangement is free. The Weyl subarrangements of type $A_{\ell}$ are represented by simple graphs. Stanley gave a characterization of freeness for this type of arrangements in terms of thier graph. In addition, The Weyl subarrangements of type $B_{\ell}$ can be represented by signed graphs. A characterization of freeness for them is not known. However, characterizations of freeness for a few restricted classes are known. For instance, Edelman and Reiner characterized the freeness of the arrangements between type $ A_{\ell-1} $ and type $ B_{\ell} $. In this paper, we give a characterization of the freeness and supersolvability of the Weyl subarrangements of type $B_{\ell}$ under certain assumption., Comment: 30 pages

The fundamental group of the complement of a hyperplane arrangement in a complex vector space is an important topological invariant. The third rank of successive quotients in the lower central series of the fundamental group was called Falk invariant of the arrangement since Falk gave the first formula and asked to give a combinatorial interpretation. In this article, we give a combinatorial formula for the Falk invariant of a signed graphic arrangement that do not have a $B_2$ as sub-arrangement.

In this paper we want to revive the object sectional matrix which encodes the Hilbert functions of successive hyperplane sections of a homogeneous ideal. We translate and/or reprove recent results in this language. Moreover, some new results are shown about their maximal growth and Persistence Theorem, a gen- eralization of Gotzmann's persistence Theorem. This suggests that further investigation of this object might cast a new light in the study of geometric consequences of maximal growth of the Hilbert function.

In this paper we give a very natural description of the bijections between the minimal CW-complex homotopy equivalent to the complement of a supersolvable arrangement $\mathcal{A}$, the $\textbf{nbc}$ basis of the Orlik-Solomon algebra associated to $\mathcal{A}$ and the set of chambers of $\mathcal{A}$. We use these bijections to get results on the first (co)homology group of the Milnor fiber of $\mathcal{A}$ and to describe a bijection between the symmetric group and the $\textbf{nbc}$ basis of the braid arrangement.

The resonant band is a useful notion for the computation of the nontrivial monodromy eigenspaces of the Milnor fiber of a real line arrangement. In this article, we develop the resonant band description for the cohomology of the Aomoto complex. As an application, we prove that real 4-nets do not exist., Comment: 23 pages, 7 figures

A rank one local system on the complement of a hyperplane arrangement is said to be admissible if it satisfies certain non-positivity condition at every resonant edges. It is known that the cohomology of admissible local system can be computed combinatorially. In this paper, we study the structure of the set of all non-admissible local systems in the character torus. We prove that the set of non-admissible local systems forms a union of subtori. The relations with characteristic varieties are also discussed., Comment: 15 pages, 3 figures

We investigate deformations of free and linear free divisors. We introduce a complex similar to the de Rham complex whose cohomology calculates deformation spaces. This cohomology turns out to be zero for many linear free divisors and to be constructible in many cases, Comment: This is the final version of the article that will appear in the Annales de l'Institut Fourier, Volume 63, 2013