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Our graph is defined by the following procedure: take $K_3\square P_3$, remove the edges of the middle triangle, add a new vertex in the middle and connect it to all three $2$-vertices.

Does it have a name? Or at least a way to write it down compactly so that a mathematician can read it (unlike 'IhMCI?TAo'). I tried looking here, but couldn't find it.

Constructed graph

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    This is just one of 19 connected cubic graphs on 10 vertices. I don't think every one of them has own name. But you described it well enough, just give it some name in you work. – 2017-02-14
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    I guess I will do that, yes. I seriously thought the graph is nice enough to have an universal name, apart from its cool properties. – 2017-02-15
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    @MatjažKrnc I can't find a name for it, but it is a simple extension of a cubic graph I would call 'cuneane' – 2017-02-15

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