Hard links in Linux are directory entries that refer to the same inode number within the same filesystem. What is the minimum link count for a directory name on a Linux system?

Hard links in Linux are directory entries that refer to the same inode number within the same filesystem. What is the minimum link count for a directory name on a Linux system?

• 4 The directory, a hard link to itself, a hard link to its parent, and hard link from its parent.
• 3 The directory name itself, a hard link to itself named " ." and the hard link ".." to its parent.
• 0 Because users cannot hard link additional names to a directory name.
• 2 The directory name itself, and a hard link to itself (named "."), within the directory. 

EXPLANATION

Users cannot hard link directories:

$ mkdir This_dir
$ ln This_dir That_dir
ln: ‘This_dir’: hard link not allowed for directory

But the system makes two hard links within a directory when it is created.  One of these is named "." and it is a link to the newly created directory itself, so the new directory will have a link count of 2, which is the minimum link count for a directory.
The other is named ".." and it is a link to  the parent directory, and will add to the parent's link count.

$ ls -ild Link.test    ## list "i"node, "l"ong list, for the "d"irectory itself.
ls: cannot access Link.test: No such file or directory
 
$ mkdir Link.test      ## make the directory.

$ ls -ild Link.test   
2238981 drwxr-xr-x 2 dan users 4096 Dec 26 13:53 Link.test
$ #Link count......^ The newly created directory has 2 hard links.
 
$ ls -ild "Link.test"  "Link.test/."  #Quoting not required, but more readable
2238981 drwxr-xr-x 2 dan users 4096 Dec 26 13:53 Link.test
2238981 drwxr-xr-x 2 dan users 4096 Dec 26 13:53 Link.test/.

When making a new directory, the system will also create a link to the parent directory named  ".."
So every time a directory is created, the link count of the parent directory increases by 1.

$ pwd
/home/dan/Test.link.dir




$ ## List inode numbers in a long listing, directory only, for "Link.test" and "Link.test/."

$ ls -ild "Link.test"   "Link.test/."
2238981 drwxr-xr-x 2 dan users 4096 Dec 26 13:53 Link.test
2238981 drwxr-xr-x 2 dan users 4096 Dec 26 13:53 Link.test/.
$ #................^ Link count is 2 for each entry and inode numbers are identical, 2238981.

$ mkdir Link.test/subdir  ## make a subdirectory under "Link.test".

$ ## Now check the link count, including the newly created subdirectory's parent, " .. "

$ ls -ild   "Link.test/"   "Link.test/."   "Link.test/subdir/.."
2238981 drwxr-xr-x 3 dan users 4096 Dec 26 15:55 Link.test/
2238981 drwxr-xr-x 3 dan users 4096 Dec 26 15:55 Link.test/.
2238981 drwxr-xr-x 3 dan users 4096 Dec 26 15:55 Link.test/subdir/..
$ #................^ Link count has increased from 2 to 3 after creating "subdir".
$ #+ The 3 names above, "Link.test", "Link.test/." and "Link.test/subdir/.." reference
$ #+ the same inode, 2238981.  

c.f.  https://www.tldp.org/LDP/gs/node5.html
"... a directory is actually just a file containing information about link-to-inode associations. Also, every directory contains at least two hard links: ``.'' (a link pointing to itself), and ``..'' (a link pointing to the parent directory)."
The root directory of a filesystem does not have a parent directory, so "/" and  "/."  and "/.."  all reference the same inode.  That is, the parent directory of root, "/..", is  "/"
E.g.

$ ls -ild  "/"   "/."   "/.."
2 drwxr-xr-x 32 root root 4096 Oct 26 13:54 /
2 drwxr-xr-x 32 root root 4096 Oct 26 13:54 /.
2 drwxr-xr-x 32 root root 4096 Oct 26 13:54 /..
$ #..........^^ Note that this link count also includes parent references (..) from subdirectories. 

See also:  http://teaching.idallen.com/cst8207/13w/notes/notes/450_file_system.html

SOURCE

https://www.tldp.org/LDP/gs/node5.html