Distance domination versus iterated domination

AbstractA k-dominating set in a graph G is a set S of vertices such that every vertex of G is at distance at most k from some vertex of S. Given a class D of finite simple graphs closed under connected induced subgraphs, we completely characterize those graphs G in which every connected induced subgraph has a connected k-dominating subgraph isomorphic to some D∈D. We apply this result to prove that the class of graphs hereditarily D-dominated within distance k is the same as the one obtained by iteratively taking the class of graphs hereditarily dominated by the previous class in the iteration chain. This strong relation does not remain valid if the initial hereditary restriction on D is dropped

Dendrites, Topological Graphs, and 2-Dominance

For each positive ordinal α, the reflexive and transitive binary relation of α-dominance between compacta was first defined in our paper [Mapping properties of co-existentially closed continua, Houston J. Math., 31 (2005), 1047-1063] using the ultracopower construction. Here we consider the important special case α =2, and show that any Peano compactum 2-dominated by a dendrite is itself a dendrite (with the same being true for topological graphs and trees). We also characterize the topological graphs that 2-dominate arcs (resp., simple closed curves) as those that have cut points of order 2 (resp., those that are not trees)

On the First-order Expressibility of Lattice Properties to Unicoherence in Continua

Many properties of compacta have “textbook” definitions which are phrased in lattice-theoretic terms that, ostensibly, apply only to the full closed-set lattice of a space. We provide a simple criterion for identifying such definitions that may be paraphrased in terms that apply to all lattice bases of the space, thereby making model-theoretic tools available to study the defined properties. In this note we are primarily interested in properties of continua related to unicoherence; i.e., properties that speak to the existence of “holes” in a continuum and in certain of its subcontinua

On Equidomination in Graphs

A graph G=(V,E) is called equidominating if there exists a value t in IN and a weight function w : V -> IN such that the total weight of a subset D of V is equal to t if and only if D is a minimal dominating set. Further, w is called an equidominating function, t a target value and the pair (w,t) an equidominating structure. To decide whether a given graph is equidominating is referred to as the EQUIDOMINATION problem. First, we examine several results on standard graph classes and operations with respect to equidomination. Furthermore, we characterize hereditarily equidominating graphs. These are the graphs whose every induced subgraph is equidominating. For those graphs, we give a finite forbidden induced subgraph characterization and a structural decomposition. Using this decomposition, we state a polynomial time algorithm that recognizes hereditarily equidominating graphs. We introduce two parameterized versions of the EQUIDOMINATION problem: the k-EQUIDOMINATION problem and the TARGET-t EQUIDOMINATION problem. For k in IN, a graph is called k-equidominating if we can identify the minimal dominating sets using only weights from 1 to k. In other words, if an equidominating function with co-domain {1,...,k} exists. For t in IN, a graph is said to be target-t equidominating if there is an equidominating structure with target value t. For both parameterized problems we prove fixed-parameter tractability. The first step for this is to achieve the so-called pseudo class partition, which coarsens the twin partition. It is founded on the requirement that vertices from different blocks of the partition cannot have equal weights in any equidominating structure. Based on the pseudo class partition, we state an XP algorithm for the parameterized versions of the EQUIDOMINATION problem. The second step is the examination of three reduction rules - each of them concerning a specific type of block of the pseudo class partition - which we use to construct problem kernels. The sizes of the kernels are bounded by a function depending only on the respective parameter. By applying the XP algorithm to the kernels, we achieve FPT algorithms. The concept of equidomination was introduced nearly 40 years ago, but hardly any investigations exist. With this thesis, we want to fill that gap. We may lay the foundation for further research on equidomination

Subsets of ωω and the Fréchet-Urysohn and αi-properties

AbstractArhangel'skiǐ defined a number of related properties called αi (i = 1, 2, 3, 4) having to do with amalgamating countably many sequences each converging to the same point. Here we use the set ωω of functions to produce examples of Fréchet spaces in the various classes and to study the relationships between the classes. We also introduce an intermediate class α1.5. Under various set-theoretic hypotheses we produce a countable Fréchet α1-space that is not first countable, and several that are α2 but not α1, including one which is α1.5 and another which is not. It is now known to be consistent that none of these kinds of spaces exist, but we also construct a countable Fréchet-Urysohn α2-space that is not first countable using only ZFC.The existence of an α2-space which is not α1 in any given model of set theory is reduced to the existence of a certain kind of space whose underlying set is (ω × ω)∪∞, with neighborhoods of ∞ defined using graphs of partial functions. Alan Dow has recently shown that every α2-space is α1 in the Laver model. A proof using the reduction theorem is outlined here and the result is used to obtain other information about this model.An example of a countable α2-topological group that is not first countable is given, and it is shown to be Fréchet-Urysohn under the relatively mild assumption p = b, as is a related separable nonmetrizable topological vector space

On what I do not understand (and have something to say): Part I

This is a non-standard paper, containing some problems in set theory I have in various degrees been interested in. Sometimes with a discussion on what I have to say; sometimes, of what makes them interesting to me, sometimes the problems are presented with a discussion of how I have tried to solve them, and sometimes with failed tries, anecdote and opinion. So the discussion is quite personal, in other words, egocentric and somewhat accidental. As we discuss many problems, history and side references are erratic, usually kept at a minimum (``see ... '' means: see the references there and possibly the paper itself). The base were lectures in Rutgers Fall'97 and reflect my knowledge then. The other half, concentrating on model theory, will subsequently appear

Bipartite graphs with close domination and k-domination numbers

Let $k$ be a positive integer and let $G$ be a graph with vertex set $V(G)$. A subset $D \subseteq V(G)$ is a $k$-dominating set if every vertex outside $D$ is adjacent to at least $k$ vertices in $D$. The $k$-domination number $\gamma_k(G)$ is the minimum cardinality of a $k$-dominating set in $G$. For any graph $G$, we know that $\gamma_k(G) \geq \gamma(G)+k-2$ where $\Delta(G)\geq k\geq 2$ and this bound is sharp for every $k\geq 2$. In this paper, we characterize bipartite graphs satisfying the equality for $k\geq 3$ and present a necessary and sufficient condition for a bipartite graph to satisfy the equality hereditarily when $k=3$. We also prove that the problem of deciding whether a graph satisfies the given equality is NP-hard in general

The complexity of connected domination and total domination by restricted induced graphs

Given a graph class C, it is natural to ask whether a given graph has a connected or a total dominating set inducing a graph of C and, if so, what is the minimal size of such a set. We give a sufficient condition on C for the intractability of this problem. This condition is fulfilled by a wide range of graph classes