More on the DVI on the mac mini.
http://www.datapro.net/techinfo/dvi_info.html#Page05
At this link you can learn about the various types of DVI connectors. It appears that the Mac Mini is a DVI-I Dual link. (three rows of eight pins and four contracts around the blade). A DVI-D link is two sets of nine pins and a solitary flat blade.
It may be possible, haven't looked into it yet, that a DVI-I dual link can transition from a DVI-I male to a DVI-D single link female (if there is such a combination?) and then couple with a DVI-D to HDMI cable. Maybe that would trick the TV if the DVI-I to HDMI doesn't work right. ???? this is conjecture.