Vacuum stability and naturalness in type-II seesaw Naoyuki Haba, Hiroyuki Ishida, Nobuchika Okada and Yuya Yamaguchi Eur. Phys. J. C, 76 6 (2016) 333 Published online: 16 June 2016 DOI: 10.1140/epjc/s10052-016-4180-z