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Classify all simple graphs $G$ on $n$ vertices such that the size of the minimal dominating set $\gamma(G)$ is 1.

Is a tree simple? This would result in many graphs having $\gamma(G)=1$.

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    Trees are simple graphs, yes, but not all trees have $\gamma(T)=1$... – 2017-02-02
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    Thank you @ParclyTaxel, how will I go about answering this question? – 2017-02-02

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The simple graphs $G$ on $n$ vertices for which $\gamma(G)=1$ are precisely the graphs with highest degree $\Delta(G)=n-1$, i.e. the graphs where one vertex $v$ is adjacent to all the others. If this is the case, $v$ alone is a dominating set; otherwise no single vertex suffices to be dominating and $\gamma(G)>1$.

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    Thank you, how will I go about finding which simple graphs have a maximal independent subset of 1? – 2017-02-02
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    @user407151 That question is answered by the complete graphs, and _only_ the complete graphs. As long as two vertices are not adjacent, they form an independent set of size grater than one. – 2017-02-02
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    Fine, but the question did not ask us to ***characterize*** those graphs, it asked us to ***classify*** them; which is essentially the same as asking us to classify all graphs. Not at all clear what sort of classification is expected. Wonder if "classify" was a typo for "characterize"? – 2017-02-02
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    @bof I presume that "characterise" was meant here with "classify". – 2017-02-02
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    The question stated "Classify" @bof – 2017-02-02