treewidth of a given graph

Question

Is the treewidth of this graph equal to 2?

I have tried to prove it through the definition of a tree decomposition.

If its not correct can someone give any hints?

Answer 1

Treewidth is always at least the clique number minus one. Your graph has a $K_4$, so its treewidth is at least 3.

Answer 2

In the survey Treewidth, partial $k$-trees, and chordal graphs by Pinar Heggernes, we have the following lemma, whose proof is given as an exercise.

Lemma. Let $(\{X_i \mid i \in I\}, T = (I, M))$ be a tree decomposition of $G = (V, E)$, and let $K \subseteq V$ be a clique in $G$. Then there exists an $i \in I$ with $K \subseteq X_i$.

Prove this lemma, and conclude that, as Yuval answered,

Corollary. For every graph $G$, $tw(G) \geq \omega(G) - 1$.

Conclude that the treewidth is at least three.

The treewidth happens to be at most three as well, but that's a different exercise.

Answer 3

The class of graphs of treewidth two is precisely the class of graphs that are $K_4$-minor-free. Because your graph contains a $K_4$-minor, it has treewidth different from two.