Eucharist and Dark Tranquillity/Septic Broiler. Also Carcass's Heartwork had quite an impact on later Gothenburg style but Eucharist and Dark Tranquillity started it.
Gothenburg Sound: The use of classic rock song structures and melodies in modern death metal, evolving in Gothenburg, Sweden, sometime in the late 80's
I've heard At The Gates and Grotesque mentioned. But....
that is not about melodic death metal... especially if we're talking about those 2 bands. It's pretty plain death metal. Listen to Gardens of Grief/In The Embrace of Evil. Basically...
once studio Fredman got involved... THEN the GOTHENBURG sound started. Trust me.. there were metal bands in that city LONG before Groutesque.. but they don't contribute to the "sound".
Fredrik Nordstrom and the first bands to use his studio are the originators. He IS a producer. Look how At The Gates changed when they went there.
So yeah... praise the bum Fredman. (Hope you get that one.)