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{\bf Xueliang Li and Yan Liu}
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{\bf Nonexistence of Triples of Nonisomorphic Connected Graphs with Isomorphic Connected $P_3$-graphs}
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In the paper ``Broersma and Hoede, {\it Path graphs}, J. Graph Theory
{\bf 13} (1989) 427-444", the authors asked a problem whether there is
a triple of mutually nonisomorphic connected graphs which have an
isomorphic connected $P_3$-graph. In this paper, we show that there is
no such triple, and thus completely solve this problem.



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