Abstract: In [3] the authors studied a topologically contractible irreducible algebraic curve C on a Q-homology plane S with logarithmic Kodaira dimension -∞.They determined such a pair (S, C) when κ(S \ C) 0 and C is smooth (cf.[3, Theorem 1.1, DOI: 10.1007/s00031-005-1121-6]). Theorem 1.1 of [3] was also proved jointly by Gurjar, Masuda, Miyanishi and Russell [2].See [6] for the strengthening results in [2] and [3].After publishing [3], the authors found a serious mistake in [3, Sec.3].More precisely, [3, Lemma 3.3] is wrong.So there is a gap in the proof of [3, Theorem 1.1].This paper is a correction to [3, Sec.3].We use the same notations and definitions as in [3].We give some results on smooth affine surfaces with κ 0 and p g = e = 0, where e denotes the topological Euler number.