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A graph with $C_3$ + 3 "ears" is called a net graph.

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In general, $C_n$ + $n$ pendant vertices are called sunlet graphs.

Is there a name for 4-sunlet graph minus two pendant vertices?

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According to ISGCI this graph is known as "A".

Looking at some references, it also seems that there is really no common well-known name.

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