I am looking at an exercise regarding decomposable graph. Above is an undirected graph. I know one necessary and sufficient condition for a decomposable graph is that any cycles of lengths $\geq4$ are chordal. Isn't it quite obvious that the circle 2-4-5-7 is not chordal?
In the answer, it only writes that the cycles 3-5-6-7 and 1-3-5-7-4-2 possess chords. Then, it suddenly jumps to the conclusion that this graph is decomposable. Don't we need to check other circles of length $\geq 4$?
Many thanks for any help!