Completely positive matrices of order $5$ with a nearly $\widehat{CP}$-graph

A particular class of graphs, called "nearly $\widehat{CP}$-graphs", extending the class of $\widehat{CP}$-graphs is introduced. From Barioli's characterization of completely positive matrices with a book-graph, two equivalent characterizations of completely positive matrices with a nearly $\widehat{CP}$-graph are deduced. By means of these results, new characterizations of completely positive matrices of order $5$ with a book-graph and some alternative demonstrations of known results are derived. A new characterization of completely positive matrices of order $5$ with a $\widehat{CP}$-graph and a new elementary proof of the main result obtained by Cedolin-Salce, regarding this particular class of matrices, are shown. It is also attempted to clarify some results obtained by Xu on completely positive matrices of order $5$, and it is shown by a counterexample that one of them is incorrect.