Hypo-efficient domination and hypo-unique domination
| Autor: | V. Samodivkin |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2016 |
| Předmět: | |
| Zdroj: | Communications in Combinatorics and Optimization, Vol 1, Iss 2, Pp 103-116 (2016) |
| Druh dokumentu: | article |
| ISSN: | 2538-2128 2538-2136 |
| DOI: | 10.22049/CCO.2016.13553 |
| Popis: | For a graph $G$ let $\gamma (G)$ be its domination number. We define a graph G to be (i) a hypo-efficient domination graph (or a hypo-$\mathcal{ED}$ graph) if $G$ has no efficient dominating set (EDS) but every graph formed by removing a single vertex from $G$ has at least one EDS, and (ii) a hypo-unique domination graph (a hypo-$\mathcal{UD}$ graph) if $G$ has at least two minimum dominating sets, but $G-v$ has a unique minimum dominating set for each $v\in V(G)$. We show that each hypo-$\mathcal{UD}$ graph $G$ of order at least $3$ is connected and $\gamma(G-v) < \gamma(G)$ for all $v \in V$. We obtain a tight upper bound on the order of a hypo-$\mathcal{P}$ graph in terms of the domination number and maximum degree of the graph, where $\mathcal{P} \in \{\mathcal{UD}, \mathcal{ED}\}$. Families of circulant graphs, which achieve these bounds, are presented. We also prove that the bondage number of any hypo-$\mathcal{UD}$ graph is not more than the minimum degree plus one. |
| Databáze: | Directory of Open Access Journals |
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