On Tutte polynomial uniqueness of twisted wheels (Q1011729)

In a \textit{fan} graph one vertex is joined to all the vertices of a path: equivalently, a \textit{fan} is formed by removing one rim edge from a \textit{wheel}; call the spokes incident with the removed rim edge \textit{external}, and the other spokes \textit{internal} spokes of the fan. Take two fans whose only intersection is one edge which is an external spoke of each, and its incident vertices, where the \textit{hubs} of the two fans are the opposite ends of that external spoke; if one joins by a new edge the two vertices of degree 2, one obtains a \textit{twisted wheel}. The \textit{Tutte polynomial} is the polynomial which Tutte [\textit{W. T. Tutte}, ``A contribution to the theory of chromatic polynomials'', Can. J. Math. 6, 80--91 (1954; Zbl 0055.17101)] called the \textit{dichromate}; a graph is \textit{\(T\)-unique} if no other graph (up to isomorphism) has the same Tutte polynomial. From the authors' abstract: ``Recently there has been much interest in determining \(T\)-unique graphs and matroids. For example, \textit{A. de Mier} and \textit{M. Noy} [``On graphs determined by their Tutte polynomials'', Graphs Comb. 20, No.\,1, 105--119 (2004; Zbl 1053.05057)] showed that wheels, ladders, Möbius ladders, square of cycles, hypercubes, and certain class of line graphs are all \(T\)-unique. In this paper we prove that the twisted wheels are also \(T\)-unique.''