Related papers: The Minimum Generating Set Problem

Let G be a finite group and S a subset of G\{0}. We call S an additive basis of G if every element of G can be expressed as a sum over a nonempty subset in some order. Let cr(G) be the smallest integer t such that every subset of G\{0} of…

In a vertex-colored graph $G = (V, E)$, a subset $S \subseteq V$ is said to be consistent if every vertex has a nearest neighbor in $S$ with the same color. The problem of computing a minimum cardinality consistent subset of a graph is…

A non-redundant integer cone generator (NICG) of dimension $d$ is a set $S$ of vectors from $\{0,1\}^d$ whose vector sum cannot be generated as a positive integer linear combination of a proper subset of $S$. The largest possible…

Given a connected graph $G=(V,E)$, a set $S\subseteq V$ is a $k$-metric generator for $G$ if for any two different vertices $u,v\in V$, there exist at least $k$ vertices $w_1,...,w_k\in S$ such that $d_G(u,w_i)\ne d_G(v,w_i)$ for every…

This paper introduces the concept of a generating set for stochastic matrices -- a subset of matrices whose repeated composition generates the entire set. Understanding such generating sets requires specifying the "indivisible elements" and…

We find finite, reasonably small, generator sets of the coordinate rings of G-character varieties of finitely generated groups for all classical groups G. This result together with the method of Grobner basis gives an algorithm for…

We study the problem of finding a maximum cardinality minimal separator of a graph. This problem is known to be NP-hard even for bipartite graphs. In this paper, we strengthen this hardness by showing that for planar bipartite graphs, the…

A 2-covering for a finite group $G$ is a set of proper subgroups of $G$ such that every pair of elements of $G$ is contained in at least one subgroup in the set. The minimal number of subgroups needed to 2-cover a group $G$ is called the…

A set of vertices $S$ of a graph $G$ is a geodetic set of $G$ if every vertex $v\not\in S$ lies on a shortest path between two vertices of $S$. The minimum cardinality of a geodetic set of $G$ is the geodetic number of $G$ and it is denoted…

Let $G=(V,E)$ be a simple graph. A dominating set of $G$ is a subset $S\subseteq V$ such that every vertex not in $S$ is adjacent to at least one vertex in $S$. The cardinality of a smallest dominating set of $G$, denoted by $\gamma(G)$, is…

Consider a nonsolvable finite group G, where R(G) represents the solvable radical of G. For any element x in G, the solvabilizer of x in G, denoted by Sol_G(x), is defined as the set of all elements y in G such that the subgroup generated…

Let $P_k$ be the graded polynomial algebra $\mathbb F_2[x_1,x_2,\ldots ,x_k]$ over the prime field with two elements, $\mathbb F_2$, with the degree of each $x_i$ being 1. We study the hit problem, set up by Frank Peterson, of finding a…

In combinatorial group testing (CGT), the objective is to identify the set of at most $d$ defective items from a pool of $n$ items using as few tests as possible. The celebrated result for the CGT problem is that the number of tests $t$ can…

Let $\Gamma(G)$ be the Gruenberg-Kegel graph of a finite group $G$. We prove that if $G$ is solvable and $\sigma$ is a cut-set for $\Gamma(G)$, then $G$ has a $\sigma$-series of length $5$ whose factors are controlled. As a consequence, we…

Two minimal generating sets of the first syzygies of a monomial ideal are produced, given the minimal generating set of the ideal.

For a simple graph $G=(V,E)$ without any isolated vertex, a cosecure dominating set $D$ of $G$ satisfies the following two properties (i) $S$ is a dominating set of $G$, (ii) for every vertex $v \in S$ there exists a vertex $u \in V…

Let G be a group of the form G_1* ... *G_n, the free product of n subgroups, and let M be a ZG-module of the form $\bigoplus_{i=1}^n M_i \otimes_{\mathbb{Z}G_i} \mathbb{Z}G$. We shall give formulae in various situations for $d_{ZG}(M)$, the…

We consider the Minimum Coverage Kernel problem: given a set $B$ of $d$-dimensional boxes, find a subset of $B$ of minimum size covering the same region as $B$. This problem is $\mathsf{NP}$-hard, but as for many $\mathsf{NP}$-hard problems…

An automorphism group of a graph $G$ is the set of all permutations of the vertex set of $G$ that preserve adjacency and non adjacency of vertices in a graph. A fixing set of a graph $G$ is a subset of vertices of $G$ such that only the…

We investigate the extent to which the exchange relation holds in finite groups $G$. We define a new equivalence relation $\equiv_{\mathrm{m}}$, where two elements are equivalent if each can be substituted for the other in any generating…