scholarly journals An Algebraic Exploration of Dominating Sets and Vizing's Conjecture

10.37236/2194 ◽  
2012 ◽  
Vol 19 (2) ◽  
Author(s):  
Susan Margulies ◽  
I. V. Hicks
Keyword(s):  
Dominating Set ◽  
Hamiltonian Path ◽  
Independent Set ◽  
Combinatorial Problems ◽  
Polynomial Equations ◽  
Dominating Sets ◽  
Critical Graphs ◽  
System Of Polynomial Equations ◽  
High Degree ◽  
Systems Of Polynomial Equations

Systems of polynomial equations are commonly used to model combinatorial problems such as independent set, graph coloring, Hamiltonian path, and others. We formulate the dominating set problem as a system of polynomial equations in two different ways: first, as a single, high-degree polynomial, and second as a collection of polynomials based on the complements of domination-critical graphs. We then provide a sufficient criterion for demonstrating that a particular ideal representation is already the universal Grobner bases of an ideal, and show that the second representation of the dominating set ideal in terms of domination-critical graphs is the universal Grobner basis for that ideal. We also present the first algebraic formulation of Vizing's conjecture, and discuss the theoretical and computational ramifications to this conjecture when using either of the two dominating set representations described above.

scholarly journals Bivariate systems of polynomial equations with roots of high multiplicity

2021 ◽  
Author(s):  
I. Nikitin
Keyword(s):  
High Multiplicity ◽  
Polynomial Equations ◽  
Support Sets ◽  
Multiplicity Formula ◽  
System Of Polynomial Equations ◽  
Systems Of Polynomial Equations

Given a bivariate system of polynomial equations with fixed support sets [Formula: see text] it is natural to ask which multiplicities its solutions can have. We prove that there exists a system with a solution of multiplicity [Formula: see text] for all [Formula: see text] in the range [Formula: see text], where [Formula: see text] is the set of all integral vectors that shift B to a subset of [Formula: see text]. As an application, we classify all pairs [Formula: see text] such that the system supported at [Formula: see text] does not have a solution of multiplicity higher than [Formula: see text].

On Dominating Sets and Independent Sets of Graphs

1999 ◽  
Vol 8 (6) ◽  
pp. 547-553 ◽  
Author(s):  
JOCHEN HARANT ◽  
ANJA PRUCHNEWSKI ◽  
MARGIT VOIGT
Keyword(s):  
Upper Bound ◽  
Maximum Degree ◽  
Dominating Set ◽  
Domination Number ◽  
Independent Set ◽  
Dominating Sets ◽  
Integral Vector ◽  
Vertex Set ◽  
Usual Concept ◽  
Dimensional Cube

For a graph G on vertex set V = {1, …, n} let k = (k1, …, kn) be an integral vector such that 1 [les ] ki [les ] di for i ∈ V, where di is the degree of the vertex i in G. A k-dominating set is a set Dk ⊆ V such that every vertex i ∈ V[setmn ]Dk has at least ki neighbours in Dk. The k-domination number γk(G) of G is the cardinality of a smallest k-dominating set of G.For k1 = · · · = kn = 1, k-domination corresponds to the usual concept of domination. Our approach yields an improvement of an upper bound for the domination number found by N. Alon and J. H. Spencer.If ki = di for i = 1, …, n, then the notion of k-dominating set corresponds to the complement of an independent set. A function fk(p) is defined, and it will be proved that γk(G) = min fk(p), where the minimum is taken over the n-dimensional cube Cn = {p = (p1, …, pn) [mid ] pi ∈ ℝ, 0 [les ] pi [les ] 1, i = 1, …, n}. An [Oscr ](Δ22Δn-algorithm is presented, where Δ is the maximum degree of G, with INPUT: p ∈ Cn and OUTPUT: a k-dominating set Dk of G with [mid ]Dk[mid ][les ]fk(p).

scholarly journals SOLVABILITY OF SYSTEMS OF POLYNOMIAL EQUATIONS OVER FINITE ALGEBRAS

2007 ◽  
Vol 17 (04) ◽  
pp. 821-835 ◽  
Author(s):  
LÁSZLÓ ZÁDORI
Keyword(s):  
Identity Element ◽  
Finite Algebra ◽  
Algorithmic Complexity ◽  
Polynomial Equations ◽  
Finite Algebras ◽  
Dichotomy Theorem ◽  
Maltsev Condition ◽  
System Of Polynomial Equations ◽  
Systems Of Polynomial Equations

We study the algorithmic complexity of determining whether a system of polynomial equations over a finite algebra admits a solution. We prove that the problem has a dichotomy in the class of finite groupoids with an identity element. By developing the underlying idea further, we present a dichotomy theorem in the class of finite algebras that admit a non-trivial idempotent Maltsev condition. This is a substantial extension of most of the earlier results on the topic.

scholarly journals Recognizing Graph Theoretic Properties with Polynomial Ideals

10.37236/386 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Jesús A. De Loera ◽  
Christopher J. Hillar ◽  
Peter N. Malkin ◽  
Mohamed Omar
Keyword(s):  
Algebraic Geometry ◽  
Convex Programming ◽  
Linear Algebra ◽  
Finite Fields ◽  
Combinatorial Problems ◽  
Polynomial Method ◽  
Polynomial Equations ◽  
Graph Theoretic ◽  
Polynomial Ideals ◽  
System Of Polynomial Equations

Many hard combinatorial problems can be modeled by a system of polynomial equations. N. Alon coined the term polynomial method to describe the use of nonlinear polynomials when solving combinatorial problems. We continue the exploration of the polynomial method and show how the algorithmic theory of polynomial ideals can be used to detect $k$-colorability, unique Hamiltonicity, and automorphism rigidity of graphs. Our techniques are diverse and involve Nullstellensatz certificates, linear algebra over finite fields, Gröbner bases, toric algebra, convex programming, and real algebraic geometry.

scholarly journals Expressing Combinatorial Problems by Systems of Polynomial Equations and Hilbert's Nullstellensatz

2009 ◽  
Vol 18 (4) ◽  
pp. 551-582 ◽  
Author(s):  
J. A. LOERA ◽  
J. LEE ◽  
S. MARGULIES ◽  
S. ONN
Keyword(s):  
Minimum Degree ◽  
Polynomial System ◽  
Combinatorial Problem ◽  
Combinatorial Problems ◽  
Stable Set ◽  
Polynomial Equations ◽  
Stability Number ◽  
The Stability ◽  
Systems Of Polynomial Equations ◽  
Hilbert Nullstellensatz

Systems of polynomial equations over the complex or real numbers can be used to model combinatorial problems. In this way, a combinatorial problem is feasible (e.g., a graph is 3-colourable, Hamiltonian, etc.) if and only if a related system of polynomial equations has a solution.For an infeasible polynomial system, the (complex) Hilbert Nullstellensatz gives a certificate that the associated combinatorial problem is infeasible. Thus, unless P = NP, there must exist an infinite sequence of infeasible instances of each hard combinatorial problem for which the minimum degree of a Hilbert Nullstellensatz certificate of the associated polynomial system grows.In the first part of the paper, we show that the minimum degree of a Nullstellensatz certificate for the non-existence of a stable set of size greater than the stability number of the graph is the stability number of the graph. Moreover, such a certificate contains at least one term per stable set of G. In contrast, for non-3-colourability, we proved that the minimum degree of a Nullstellensatz certificate is at least four. Our efforts so far have only yielded graphs with Nullstellensatz certificates of precisely that degree.In the second part of this paper, for the purpose of computation, we construct new polynomial encodings for the problems of finding in a graph its longest cycle, the largest planar subgraph, the edge-chromatic number, or the largest k-colourable subgraph. We include some applications to graph theory.

scholarly journals On Independent Transversal Dominating Sets in Graphs

2021 ◽  
Vol 14 (1) ◽  
pp. 149-163
Author(s):  
Daven Sapitanan Sevilleno ◽  
Ferdinand P. Jamil
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Independent Set ◽  
Maximum Independent Set ◽  
Total Domination ◽  
Dominating Sets ◽  
Total Domination Number ◽  
Minimum Cardinality ◽  
Total Dominating Set ◽  
Positive Integers

A set S ⊆ V (G) is an independent transversal dominating set of a graph G if S is a dominating set of G and intersects every maximum independent set of G. An independent transversal dominating set which is a total dominating set is an independent transversal total dominating set. The minimum cardinality γit(G) (resp. γitt(G)) of an independent transversal dominating set (resp. independent transversal total dominating set) of G is the independent transversal domination number (resp. independent transversal total domination number) of G. In this paper, we show that for every positive integers a and b with 5 ≤ a ≤ b ≤ 2a − 2, there exists a connected graph G for which γit(G) = a and γitt(G) = b. We also study these two concepts in graphs which are the join, corona or composition of graphs.

scholarly journals Expressing Combinatorial Problems by Systems of Polynomial Equations and Hilbert's Nullstellensatz – Corrigendum

2009 ◽  
Vol 19 (1) ◽  
pp. 159-159
Keyword(s):  
Combinatorial Problems ◽  
Polynomial Equations ◽  
Cambridge University ◽  
Systems Of Polynomial Equations

On page 551 the first author name should be J. A. De Loera (NOT J. A Loera)The correct citation for this paper isJ. A. De Loera, J. Lee, S. Margulies and S. Onn (2009) Expressing Combinatorial Problems by Systems of Polynomial Equations and Hilbert's Nullstellensatz. Combinatorics, Probability and Computing, 18 (4) July, 551–582 doi:10.1017/S0963548309009894, Published online by Cambridge University Press 28 April 2009.

scholarly journals Some Results on Equi Independent Equitable Dominating Sets in Graph

2015 ◽  
Vol 7 (3) ◽  
pp. 77-85 ◽  
Author(s):  
S. K. Vaidya ◽  
N. J. Kothari
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Independent Set ◽  
Dominating Sets ◽  
Minimum Cardinality

A subset D of V(G) is called an equitable dominating set if for every ? ? V(G) - D, there exists a vertex u ? D such that u? ? E(G) and | deg(u) – deg(?) | ? 1. A subset D of V(G) is called an equitable independent set if for any D of V(G), ? ? Ne(u) for all ? ? D - {u}where, Ne(u) = {? ? V(G)/? ? N(u), | deg(u) – deg(?) | ? 1}. An equitable dominating set D is said to be an equi independent equitable dominating set if it is also an equitable independent set. The minimum cardinality of an equi independent equitable dominating set is called equi independent equitable domination number which is denoted by ie. We investigated an equi independent equitable domination number for some special graphs.

Construction of systems of polynomial equations with exact p-Divisibility via the covering method

2014 ◽  
Vol 13 (06) ◽  
pp. 1450013 ◽  
Author(s):  
Francis N. Castro ◽  
Ivelisse M. Rubio
Keyword(s):  
Exponential Sums ◽  
Polynomial Equations ◽  
Prime Field ◽  
Elementary Method ◽  
Covering Method ◽  
Systems Of Polynomial Equations

We present an elementary method to compute the exact p-divisibility of exponential sums of systems of polynomial equations over the prime field. Our results extend results by Carlitz and provide concrete and simple conditions to construct families of polynomial equations that are solvable over the prime field.

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