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Hypo-efficient domination and hypo-unique domination | ||
| Communications in Combinatorics and Optimization | ||
| مقاله 2، دوره 1، شماره 2، اسفند 2016، صفحه 103-116 اصل مقاله (517.94 K) | ||
| نوع مقاله: Original paper | ||
| شناسه دیجیتال (DOI): 10.22049/cco.2016.13553 | ||
| نویسنده | ||
| Vladimir Samodivkin* | ||
| University of Architecture, Civil Еngineering and Geodesy; Department of Mathematics | ||
| چکیده | ||
| 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. | ||
| کلیدواژهها | ||
| Domination number؛ Efficient Domination؛ unique domination؛ hypo-property | ||
| مراجع | ||
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