What is $C_4$ + two "ears" called? A graph with $C_3$ + 3 "ears" is called a net graph. 



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

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