Authors: Andrew Goodall ¹; Criel Merino ²; Anna de Mier ^3,4; Marc Noy ^3,4

• 1 Department of Applied Mathematics and Institute of Theoretical Computer Science
• 2 Instituto de Matematicas
• 3 Universitat Politècnica de Catalunya [Barcelona]
• 4 Universitat Politècnica de Catalunya = Université polytechnique de Catalogne [Barcelona] [UPC]

C. Merino [Electron. J. Combin. 15 (2008)] showed that the Tutte polynomial of a complete graph satisfies $t(K_{n+2};2,-1)=t(K_n;1,-1)$. We first give a bijective proof of this identity based on the relationship between the Tutte polynomial and the inversion polynomial for trees. Next we move to our main result, a sufficient condition for a graph G to have two vertices u and v such that $t(G;2,-1)=t(G-\{u,v\};1,-1)$; the condition is satisfied in particular by the class of threshold graphs. Finally, we give a formula for the evaluation of $t(K_{n,m};2,-1)$ involving up-down permutations.