I am asked to find the Tutte polynomial of $K_4 − e$ by definition of Tutte polynomial and delete-contract definition respectively.
I have calculated and i get
$K_4 = x^3 + 3x^2 + 2x + 4xy + 2y + 3y^2 + y^3$
which should be correct.
But how can i relate that with $K_4 − e$?
And for the second bit with the delete-contract definition bit, from my lecture note i get
polynomial of $K_3$ = polynomial of $K_3-e$ + polynomial of $K_3/e$.
Are there any way i can find the polynomial of $K_3/e$?
Thanks
direct definition
Definition. For an undirected graph $G=(V, E)$ one may define the Tutte polynomial as $$ T_G(x, y)=\sum_{A \subseteq E}(x-1)^{k(A)-k(E)}(y-1)^{k(A)+|A|-|V|}, $$ where $k(A)$ denotes the number of connected components of the graph $(V, A)$. In this definition it is clear that $T_G$ is well-defined and a polynomial in $x$ and $y$.
delete-contract definition
From the wikipedia,
The deletion-contraction recurrence for the Tutte polynomial, $$ T_G(x, y)=T_{G \backslash e}(x, y)+T_{G / e}(x, y), \quad e \text { not a loop nor a bridge. } $$
watch out! where $e$ can not be a loop nor a bridge.
I only plot part of the tree, but that's enough. In the large tree,the Tutte polynomial of a node $V$ is 【sum of all of the Tutte polynomials of the leaves given that the node $V$ being set as root】