Restoring /root and /home by dd

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The short answer is: none
Because, as you correctly denoted:

So i thin[k] the right way is to write them straight to [/dev/sdaXX.]

But in consequence you must also backup the whole /dev/sdaXX partition (XX = 79 on an Xperia 10 III) in the first place, which contains both, the /home and /root volumes (and a little LVM metadata).
So, if you have a chance to perform the backup again, do that as described by @wetab73 here (some other variants but also wrong guesses are described further down in this discussion thread).

But maybe I am missing the point, because these statements are not comprehensible for me:

I tried to write it back to the /dev/mapper/root and /dev/mapper/home - but i got a broken /home Partition. Something with size is wrong.

  • Why do you want to restore a “broken” home volume, and what do you exactly mean by “broken”?
    If the filesystem is broken, backup as much as you can from the mounted filesystem (e.g. by using tar, maybe with the GNU tar option --ignore-failed-read) and then use fsck (starting with a “dry run” by -N) to repair the filesystem.

  • Something with size is wrong.

    • Which command line emitted which error message (please copy&paste)?
    • Did you recreate the /root and /home volumes (if so: how?), or how come that a volume size is different now?
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