Gerd Fricke

• PhD
• Morehead State University

We prove that for any tree T, γ ≤2 (T)≤γ γ (T)≤ir(T)≤γ(T), where γ ≤2 (G) is the distance-2 domination number, ir(T) is the (lower) irredundance number, γ(T) is the domination number, and γ γ (T), newly defined here, equals the minimum cardinality of a set of vertices that dominates a minimum dominating set of T.

A set S⊆V is a dominating set of a graph G=(V,E) if every vertex in V-S is adjacent to at least one vertex in S. The domination number γ(G) of G equals the minimum cardinality of a dominating set S in G; we say that such a set S is a γ-set. In this paper we consider the family of all γ-sets in a graph G and we define the γ-graph G(γ)=(V(γ),E(γ)) of...

Most of the research on domination focuses on vertices dominating other vertices. In this paper we consider vertex-edge domination where a vertex dominates the edges incident to it as well as the edges adjacent to these incident edges. The minimum cardinality of a vertex-edge dominating set of a graph G is the vertex-edge domination number γ ve (G)...

Abstract We explore various types of criticality with respect to dierentiating- dominating sets, or identifying codes. Existence and characterization re- sults are included. We conclude with open problems. Keywords: dierentiating-domination, identifying code, critical

A legal placement of Queens is any placement of Queens on an order N chessboard in which any two attacking Queens can be separated by a Pawn. The Queens independence separation number is the minimum number of Pawns which can be placed on an N × N board to result in a separated board on which a maximum of m independent Queens can be placed. We prove...

We define a legal placement of Queens to be any placement in which any two attacking Queens can be separated by a Pawn. The Queens separation number is defined to be equal to the minimum number of Pawns which can separate some legal placement of m Queens on an order n chess board. We prove that n + 1 Queens can be separated by 1 Pawn and conjecture...

A real-valued function g : V → [0, 1] on the vertex set of a graph is irredundant if g(v) > 0 implies there exists a vertex w in N[v], the closed neighborhood of v, such that g(N [w]) = 1. Irredundant functions are a real-valued generalization of characteristic functions for irredundant sets. In this paper we show that in any graph, the maximum irr...

Each king on an n×n chessboard is said to attack its own square and its neighboring squares, i.e., the nine or fewer squares within one move of the king. A set of kings is said to form an irredundant set if each attacks a square attacked by no other king in the set. We prove that the maximum size of an irredundant set of kings is bounded between (n...

A set S is an oensiv e alliance if for every vertex v in its boundary N(S) S it holds that the majority of vertices in v's closed neighbour- hood are in S. The oensiv e alliance number is the minimum cardi- nality of an oensiv e alliance. In this paper we explore the bounds on the oensiv e alliance and the strong oensiv e alliance numbers (where a...

An e = 1 function is a function f : V (G) 7! (0; 1) such that every non-isolated vertex u is adjacent to some vertex v such that f(u) + f(v) = 1, and every isolated vertex w has f(w) = 1. A theory of e = 1 functions is developed focussing on minimal and maximal e = 1 functions. Relationships are traced between e = 1 parameters and some well-known d...

Let G = (V, E) be a graph and let S ⊆ V.. The set S is a dominating set of G is every vertex of V − S is adjacent to a vertex of S. A vertex v of G is called S-perfect if |N[ν]∩S| = 1 where N[v] denotes the closed neighborhood of v. The set S is defined to be a perfect neighborhood set of G if every vertex of G is S-perfect or adjacent with an S-pe...

A real-valued function g : V → [0, 1] on the vertex set of a graph is irredundant if g(v) > 0 implies there exists a vertex w in the closed neighborhood of v, N[v], such that g(N[w]) = 1. Irredundant functions are a real-valued generalization of irredundant sets. In this paper we define a real-valued analog to the well-known graph parameter ir(G),...

Let n ⩾ 1 be an integer and let G be a graph of order p. A set D of vertices of G is a total n-dominating set of G if every vertex of V(G) is within distance n from some vertex of D other than itself. The minimum cardinality among all total n-dominating sets of G is called the total n-domination number and is denoted by γtn(G). A set S of vertices...

Let T be a tree, A its adjacency matrix, and α a scalar. We describe a linear-time algorithm for reducing the matrix αIn + A. Applications include computing the rank of A, finding a maximum matching in T, computing the rank and determinant of the associated neighborhood matrix, and computing the characteristic polynomial of A.

Let i(G) (i(G), respectively) be the independent domination number (i.e. smallest cardinality of a maximal independent vertex subset) of the p-vertex graph G (the complement G of G, respectively).We prove limp→∞[maxGi(G)i(G)/p2] = 116.

LetPm denote an equilateral polygon ofm sides with each side having length 1 and we allow the sides to cross and vertex repetitions. We consider the following question. What is the smallest widthtm of a horizontal strip in the Euclidean plane that contains aPm? This problem has its origins in Euclidean Ramsey theory. Whenm is even, it is easy to se...

For more than 250 years combinatorial problems on chessboards have been studied and published in numerous books on recreational mathematics. Two problems of this type include the problem of finding a placement of n non-attacking queens on an n×n chessboard and the problem of determining the minimum number of queens which are necessary to cover ever...

This paper investigates cases where one graph parameter, upper fractional domination, is equal to three others: independence, upper domination and upper irredundance. We show that they are all equal for a large subclass, known as strongly perfect graphs, of the class of perfect graphs. They are also equal for odd cycles and upper bound graphs. Howe...

A graph is representable modulo n if its vertices can be labeled with distinct integers between 0 and n, the difference of the labels of two vertices being relatively prime to n if and only if the vertices are adjacent. Erdős and Evans recently proved that every graph is representable modulo some positive integer. We derive a combinatorial formulat...

Let $S$ be a set of vertices in a graph $G = (V, E)$. The authors state that a vertex u in S has a private neighbor (relative to $S$) if either $u$ is not adjacent to any vertex in $S$ or $u$ is adjacent to a vertex $w$ that is not adjacent to any other vertex in $S$. Based on the notion of private neighbors, a set of eight graph theoretic paramete...

In [J. Boland, R. Laskar, C. Turner and G. Domke, ibid. 70, 131-135 (1990; Zbl 0694.05056)] the concept of a mod sum graph (MSG) is introduced. It is shown that trees, cycles, and certain bipartite graphs are MSGs and complete graphs are not MSGs. In this paper a simple observation is used to prove that certain graphs are MSGs, while some are not....

A strong matching in a graph G=(V,E) is a matching in which no two edges are joined by an edge of G. A set S of vertices in G is open- open irredundant if for every vertex u in S, a vertex v adjacent to u is not adjacent to any vertex in S-u. If β * denotes the maximum number of edges in a strong matching and OOIR denotes the maximum number of vert...

This paper deals with the familiar notions of domination, independence and irredundance in an undirected graph. The corresponding minimum, maximum, minimax and maximin parameters are described in terms of vertex subsets having the qualifying properties. Each such set is treated as a characteristic vector. This approach is then generalized by allowi...

For G=(V,E) let g:V→{0,1,2,⋯,k} and let g(S)=∑ v∈S g(v) for any set S⊆V, where |g|=g(V). Such a function g is a k-dominating function if for every v∈V, g(N[v])≥k. The k-domination number of G is given by γ k (G)=min{|g|:g is a k-dominating function of G}. A function g is a k-packing function if for every v∈V, g(N[v])≤k. The k-packing number of G is...

Let t be a positive number sequence and define the sequence space Ω(t) := {x: Xk = O (tk)}. Characterizations are given for matrices that map the spaces l1, l∞, c, or c0 into Ω(t), thus ensuring that the transformed sequence converges at least as fast as t. These results yield information about matrices that map l1, l∞, c, or c0 into G := Ur∈(0,1)...

This paper studies a nondiscrete generalization of Γ(G), the maximum cardinality of a minimal dominating set in a graph G=(V,E). In particular, a real-valued function is dominating if for each vertex υ ϵ V, the sum of the values assigned to the vertices in the closed neighborhood of υ, N[υ], is at least one, i.e., . The weight of a dominating funct...

A total dominating set of a graph G=(V,E) is a set S of vertices such that every vertex v∈V is adjacent to some vertex in S. The upper total domination number Γ t is the largest size of a minimal total dominating set. A total dominating function is the continuous analogue of the characteristic function of a total dominating set. The upper fractiona...

One of the common uses of summability theory is found in its applications to power series. A partial listing such as [l]-[5], [13] , and [15] - [17] might serve to remind us of the many instances of summability theory applied to power series. In some studies, the summability transformation is applied to the sequence of partial sums of the power ser...

A NOTE ON BOUNDED INDEX AND BOUNDED VALUE DISTRIBUTION

Suppose ∑ n = 0 ∞ a n z n has radius of convergence R and σ N ( z ) = | ∑ n = N ∞ a n z n | . Suppose | z 1 | < | z 2 | < R , and T is either z 2 or a neighborhood of z 2 . Put S = { N | σ N ( z 1 ) > σ N ( z ) for z ϵ T } . Two questions are asked: (a) can S be cofinite? (b) can S be infinite? This paper provides some answers to these questions. T...

In this paper the author shows that a well known sufficient condition for strict cycliclty of a weighted shift on ℓp is not a necessary condition for any p with 1<p<∞.

A sufficient condition for a canonical product to be of bounded index is given, from which most of the well known results can be obtained as easy corollaries. Let f be an entire function of exponential type with order p and lower order λ. If σ - λ < 1 then there exists an entire function g of bounded index such that log M(r,f) ∼ log M(r, g). This s...

This paper studies the spectral properties of a class of operators known as phase operators which originated in the study of harmonic oscillator phase. Ifantis conjectured that such operators had no point spectrum. It was later shown that certain phase operators were, in fact, absolutely continuous and that all phase operators at least had an absol...

A result of Singh and Sreenivasulu is proved under less restrictive hypothesis. It is also shown that if a condition on the coefficients is not satisfied, the theorem will not hold.

This paper examines the relationship between the concept of bounded index and the radius of univalence, respectively p-valence, of entire functions and their derivatives at arbitrary points in the plane.

This paper examines the relationship between the concept of bounded index and the radius of univalence, respectively p-valence, of entire functions and their derivatives at arbitrary points in the plane.