1 on 1 DMing

Good morning everyone! I'm new here so apologies if this post has been made before.

Ive been trying to find a dnd group for me and my wife but we have a relatively small and busy social group so I was instead wondering what it takes to do a 1 on 1 DnD game? I'm sure people have done it but how do encounters and combat work? Does the DM ever include a character for themselves to play along with the solo player so they can experience both sides of the game at once? For those that have any tips it would be greatly appreciated! Thank you!