Using Domination to Analyze RNA Structures.

Understanding RNA molecules is important to genomics research. Recently researchers at the Courant Institute of Mathematical Sciences used graph theory to model RNA molecules and provided a database of trees representing possible secondary RNA structures. In this thesis we use domination parameters to predict which trees are more likely to exist in nature as RNA structures. This approach appears to have promise in graph theory applications in genomics research

Disjoint Dominating Sets with a Perfect Matching

In this paper, we consider dominating sets $D$ and $D'$ such that $D$ and $D'$ are disjoint and there exists a perfect matching between them. Let $DD_{\textrm{m}}(G)$ denote the cardinality of smallest such sets $D, D'$ in $G$ (provided they exist, otherwise $DD_{\textrm{m}}(G) = \infty$). This concept was introduced in [Klostermeyer et al., Theory and Application of Graphs, 2017] in the context of studying a certain graph protection problem. We characterize the trees $T$ for which $DD_{\textrm{m}}(T)$ equals a certain graph protection parameter and for which $DD_{\textrm{m}}(T) = \alpha(T)$, where $\alpha(G)$ is the independence number of $G$. We also further study this parameter in graph products, e.g., by giving bounds for grid graphs, and in graphs of small independence number

A characterization of (2γ,γp)-trees

AbstractLet G=(V,E) be a graph. A set S⊆V is a dominating set of G if every vertex not in S is adjacent with some vertex in S. The domination number of G, denoted by γ(G), is the minimum cardinality of a dominating set of G. A set S⊆V is a paired-dominating set of G if S dominates V and 〈S〉 contains at least one perfect matching. The paired-domination number of G, denoted by γp(G), is the minimum cardinality of a paired-dominating set of G. In this paper, we provide a constructive characterization of those trees for which the paired-domination number is twice the domination number