Locating and paired-dominating sets in graphs

In this paper, we continue the study of paired-domination in graphs introduced by Haynes and Slater [T.W. Haynes, P.J. Slater, Paired-domination in graphs, Networks 32 (1998), 199–206]. A paired-dominating set of a graph G with no isolated vertex is a dominating set S of vertices whose induced subgraph has a perfect matching. We consider paired-dominating sets which are also locating sets, that is distinct vertices of G are dominated by distinct subsets of the paired-dominating set. We consider three variations of sets which are paired-dominating and locating sets and investigate their properties.     

Almost semirecursive sets

LetA be a subset of , and leta∉A. The setA is said to be almost semirecursive, if there is a two-place general recursive functionf such thatf(x, y)ε{x, y, a}∧({x, y}⊆A⇌f(x, y)εA) for all . Among other facts, it is proved that ifA and are almost semirecursive sets, thenA is a semirecursive set, and that there exists a wsr^*-set that is neither a wsr-nor an almost semirecursive set. Translated fromMatematicheskie Zametki, Vol. 66, No. 2, pp. 188–193, August, 1999.     

Some formal relationships among soft sets,fuzzy sets,and their extensions

We prove that every hesitant fuzzy set on a set E can be considered either a soft set over the universe $[0,1]$ or a soft set over the universe E. Concerning converse relationships, for denumerable universes we prove that any soft set can be considered even a fuzzy set. Relatedly, we demonstrate that every hesitant fuzzy soft set can be identified with a soft set, thus a formal coincidence of both notions is brought to light. Coupled with known relationships, our results prove that interval type-2 fuzzy sets and interval-valued fuzzy sets can be considered as soft sets over the universe $[0,1]$. Altogether we contribute to a more complete understanding of the relationships among various theories that capture vagueness and imprecision.     

Solution of the Cameron-Erdös problem for groups of prime order

A subset A of a group G is sum-free if a + b does not belong to A for any a, b ∈ A. Asymptotics of the number of sum-free sets in groups of prime order are proved.     

Strongly Amorphous Sets and Dual Dedekind Infinity

1. If A is strongly amorphous (i.e., all relations on A are definable), then its power set P(A) is dually Dedekind infinite, i. e., every function from P(A) onto P(A) is injective. 2. The class of “inexhaustible” sets is not closed under supersets unless AC holds.     

On completely nonmeasurable unions

Assume that there is no quasi-measurable cardinal not greater than 2^ω . We show that for a c. c. c. σ -ideal 𝕀 with a Borel base of subsets of an uncountable Polish space, if 𝒜 is a point-finite family of subsets from 𝕀, then there is a subfamily of 𝒜 whose union is completely nonmeasurable, i.e. its intersection with every non-small Borel set does not belong to the σ -field generated by Borel sets and the ideal 𝕀. This result is a generalization of the Four Poles Theorem (see [1]) and a result from [3]. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

An extension of building sets and relative difference sets

Building sets are a successful tool for constructing semi‐regular divisible difference sets and, in particular, semi‐regular relative difference sets. In this paper, we present an extension theorem for building sets under simple conditions. Some of the semi‐regular relative difference sets obtained using the extension theorem are new in the sense that their ambient groups have smaller ranks than previously known. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 50–57, 2000     

A Difference Set in \left( {\mathbb{Z}/4\mathbb{Z}} \right)^3 \times \mathbb{Z}/5\mathbb{Z}

We present a (320, 88, 24)-difference set in , the existence of which was previously open. This new difference set improves a theorem of Davis-Jedwab with the removal of the exceptional case. It also enables us to state a theorem of Schmidt on Davis-Jedwab difference sets more neatly.     

On πgs-closed sets in topological spaces

Summary A new class of sets called πgs-closed sets is introduced and its properties are studied. Moreover the notions of πgs-T_1/2 spaces and πgs-continuity are introduced.     

Sets characterized by missing sums and differences

A more sums than differences (MSTD) set is a finite subset S of the integers such that |S+S|>|S−S|. We show that the probability that a uniform random subset of {0,1,…,n} is an MSTD set approaches some limit ρ>4.28×¹⁰−4. This improves the previous result of Martin and O?Bryant that there is a lower limit of at least 2×¹⁰−7. Monte Carlo experiments suggest that ρ≈4.5×¹⁰−4. We present a deterministic algorithm that can compute ρ up to arbitrary precision. We also describe the structure of a random MSTD set S⊆{0,1,…,n}. We formalize the intuition that fringe elements are most significant, while middle elements are nearly unrestricted. For instance, the probability that any “middle” element is in S approaches 1/2 as n→∞, confirming a conjecture of Miller, Orosz, and Scheinerman. In general, our results work for any specification on the number of missing sums and the number of missing differences of S, with MSTD sets being a special case.     

Large Sets and Overlarge Sets of Triangle-Decomposition

There are six types of triangles:undirected triangle,cyclic triangle,transitive triangle,mixed-1triangle,mixed-2 triangle and mixed-3 triangle.The triangle-decompositions for the six types of triangles havealready been solved.For the first three types of triangles,their large sets have already been solved,and theiroverlarge sets have been investigated.In this paper,we establish the spectrum of LT_i(v,λ),OLT_i(v)(i=1,2),and give the existence of LT_3(v,λ)and OLT_3(v,λ)with λ even.     

Approximation of analytic sets by Nash tangents of higher order

The aim of this paper is to show that for every locally analytic subset X of C ^m and every there exist a neighborhood V of a in C ^m and a sequence of Nash subsets of V converging to such that X _ν and X satisfy a certain condition for tangency of order ν. Next it is shown that this condition implies that for sufficiently large ν the multiplicities of X _ν and X at a are equal.      

Constructions of Partial Difference Sets and Relative DifferenceSets Using Galois Rings

We use Galois rings to construct partial difference sets and relative difference sets in non-elementary abelianp-groups. As an example, we also use Galois ringG R(4, 2) to construct a (96,20,4) difference set in Z₄ × Z₄ × Z₆.Dedicated to Hanfried Lenz on the occasion of his 80th birthday     

Invariant Constructions of Simple and Maximal Sets

The main results of the present paper are the following theorems: 1. There is no e ∈ ω such that for any A, B ? ω, S^A = W is simple in A, and if A′ ?_T B′, then S^A =* S^B. 2 There is an e ∈ ω such that for any A, B ? ω, M^A = W_e is incomplete maximal in A, and if A =* B, then M^A ?_T M^B.     

Tight bounds for eternal dominating sets in graphs

The eternal domination number of a graph is the number of guards needed at vertices of the graph to defend the graph against any sequence of attacks at vertices. We consider the model in which at most one guard can move per attack and a guard can move across at most one edge to defend an attack. We prove that there are graphs G for which , where γ_∞(G) is the eternal domination number of G and α(G) is the independence number of G. This matches the upper bound proved by Klostermeyer and MacGillivray.     

Not every -set is perfectly meager in the transitive sense

We prove the following theorems:

1.

It is consistent with ZFC that there exists a - set which is not perfectly meager in the transitive sense.

2.

Every set which is perfectly meager in the transitive sense has the property.

3.

The product of two sets perfectly meager in the transitive sense has also that property.

     

On Pseudocomplemented and Stone Ordered Sets

The aim of this paper is to characterize both the pseudocomplemented and Stone ordered sets in a manner similar to that used previously for Boolean and distributive ordered sets. The sublattice G(A) of the Dedekind–Mac Neille completion DM(A) of an ordered set A generated by A is said to be the characteristic lattice of A. We will show that there are distributive pseudocomplemented ordered sets whose characteristic lattices are not pseudocomplemented. We can define a stronger notion of pseudocomplementedness by demanding that both A and G(A) be pseudocomplemented. It turns out that the two concepts are the same for finite and Stone ordered sets.     

Partition the vertices of a graph into one independent set and one acyclic set

For a given graph G, if the vertices of G can be partitioned into an independent set and an acyclic set, then we call G a near-bipartite graph. This paper studies the recognition of near-bipartite graphs. We give simple characterizations for those near-bipartite graphs having maximum degree at most 3 and those having diameter 2. We also show that the recognition of near-bipartite graphs is NP-complete even for graphs where the maximum degree is 4 or where the diameter is 4.