Find the chromatic polynomial $c_{G}(k)$ by:
relation $c_{G}(k) = c_{G−e}(k) − c_{G/e}(k)$
principle of inclusion and exclusion.
The graph looks like a triangle of vertices a (top), b (right) , and d (bottom) all connected to one another. Then vertex c is connected to vertex b. And that's it. Basically a triangle with added edge connected to one vertex of the triangle.
This is graph theory and combinatorics.Please help how to solve this.
With deletion-contraction, I would be clever about this: $c_{G}(k) = c_{G - bd}(k) - c_{G/bd}(k)$. So if we remove $bd$, we have a path on four vertices. And so, it is a tree with chromatic polynomial $k(k-1)^{3}$. Similarly, we contract the edge $bd$ to get a path on $3$ vertices, giving us the chromatic polynomial $k(k-1)^{2}$. And so $c_{G}(k) = k(k-1)^{2}(k-2)$.
I'm not familiar with the inclusion-exclusion approach. Perhaps someone who is can offer some advice on that end.