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We study the K\H{o}nig type property for non-simple polyominoes. We provethat, for closed path polyominoes, the polyomino ideals are of K\H{o}nig type,extending the results of Herzog and Hibi for simple thin polyominoes. As anapplication of this result, we give a combinatorial interpretation for thecanonical module of the coordinate ring of a sub-class of closed pathpolyominoes, namely circle closed path polyominoes. In this case, we computealso the Cohen-Macaulay type and we show that $K[\mathcal{P}]$ is a level ring.

Non-simple polyominoes of K\H{o}nig type and their canonical module