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Let $G=(V_1,E_1)$ be a simple graph with vertex set $\{v_1,v_2,\ldots,v_n\}$ and let $G'=(V_2,E_2)$ be another copy of $G$ with vertex set $\{u_1,u_2,\ldots,u_n\}$. Assume $V_1\cap V_2= \emptyset$.

Let $H=(V,E)$ be a graph with $V=V_1 \cup V_2$ and $E=E_1\cup E_2\cup \{u_1v_1，u_2v_2, \ldots, v_nv_n\}$. It is obtained from $G$ by a graph operation as above. So, is there any name of this operation?

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I dont't know of a standard name, but it is $G \square K_2$, where $\square$ denotes the Cartesian product.

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  • $\begingroup$ Not quite, surely. If mapping $u_i$ to $v_i$ is an isomorphism, then it is exactly the Cartesian product with $K_2$, but otherwise it has no particular name. $\endgroup$
    – Gordon Royle
    Commented Oct 21, 2021 at 23:11
  • $\begingroup$ Good point. I assumed the OP meant that $u_i$ is a copy of $v_i$. $\endgroup$
    – Tony Huynh
    Commented Oct 21, 2021 at 23:29
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Not necessarily a standard name but in this paper I and my coauthors called it cloning the graph.

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