I read of the following deletion contraction recurrence for Tutte polynomial of a graph: $T_G = T_{G-e} + T_{G/e}$ where $e$ is neither a loop nor a bridge. I decided to do some examples, so I checked for $G=K_3$. Taking any one edge as $e$, $G-e = P_3$ and $G/e = P_2$. But $T_{K_3}=x^2+x+y$, $T_{P_3}=x^2$, $T_{P_2}=x$. Do I need to avoid edges of type $uv$ such that $N(u)\cap N(v) \neq \emptyset$? That is the only difference I have found from the other examples I did, which worked fine. I cross-checked the computations on Wolfram, so it's not a calculation error.
I was reading this in context of relating the Tutte polynomial of hyperplane arrangements with that of graphs, and noticed the issue while working out $K_3$ and its associated graphical arrangement $A_2$. I also had issues with the similar recurrence for hyperplane arrangements in case of a non-graphical arrangement -- the Linial arrangement in $\mathbb{R}^3$ -- but this could be an error in my calculations (I am not proficient enough with Wolfram to check it).
Please help with insight/intuition.
Tag request : hyperplane-arrangements
The equation $\;C_2\!:=\! G/e\! =\! P_2\;$ is not correct. $C_2$ is a graph with two parallel edges while $P_2$ has only one. The Tutte polynomial of $\;C_2\;$ is $\;T_{C_2}\!=\!x+y.\;$ With this correct Tutte polynomial, the deletion contraction recurrence is now correct. In the Wolfram language $\;\texttt{C2=CycleGraph[2]}\;$.
There are other applications of the deletion contraction recurrence. In the Chromatic polynomial case, parallel edges can be reduced to a single edge although not for the Tutte polynomial.