Tiny Epic Game of Thrones

Can a White Walker or the Night King be in the same domain with a nonplayer house?

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No
If you add a White Walker (or the Night King) into a domain with units of a non-player house (that is not allied to you), one unit is placed onto the Terror Pool and the remaining units must retreat. This is true even if the non-player house is terrified (terrified houses are not allied to the Night King). Meanwhile if the non-player house is allied to you, then this starts a battle with you.
Related Rule(s)