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Given a graph $G$, the total graph of $G$, denoted $T(G)$, is the graph with vertex set $V(G) \cup E(G)$, where $a$ and $b$ are adjacent in $T(G)$ if and only if they are adjacent or incident in $G$.

Is there any characterization of properties of two graphs $G$ and $H$ such that $T(G)$ is isomorphic to $T(H)$?

Cross-posted at MSE.

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    $\begingroup$ I edited the question to include the definition of a total graph. Note that it is usually discouraged to cross post on both sites without giving proper time to answer. $\endgroup$
    – Tony Huynh
    Commented Jan 30, 2016 at 12:22

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It is not difficult to see that both $G$ and $L(G)$ are disjoint induced subgraphs of the graph $T(G)$. For your question, in the below paper:

[1] M. Behzad and H. Radjavi, The total group of a graph, Proc. Amer. Math. Soc. 19 (1968), 158-163. MR 36 #1358.

It is proved that, the graph $G$ is isomorphic to the graph $H$ if and only if $T(G)$ is isomorphic to $T(H)$.

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