# treewidth of a given graph

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?

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

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.

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.