Apparently, I have made a configuration changed which has degraded network discovery. I believe that it may have resulted from attempting to "eliminate" the need to enter a password when accessing a public share file on a remote computer.

I have a home LAN network of three computers. If you look at the attached image for computer3, you will see how the network appeared on all three computers.

The attached image for computer1 now shows only the icon for the "Windows Network". Furthermore, the "Windows Network" icon is very very slow to appear. Eventually, I can still access the public share files on the other computers.

Two additional points:

1. Why do I believe it is a configuration issue? Well, computer3 was never "touched". I experimented with computers 1&2 to allow accessing the remote computer without having to enter a password. That didn't work. Now these computers do not show the image as displayed for computer3 and are very slow to access the network.
2. As an attempt to resolve this issue, I removed and reinstalled all "GVFS" files for computer1, but that did not work. I also removed and reinstalled "CIFS-UTILS". Again no change.

UPDATE: I have apparently solved one issue. Issuing the command "net usershare info" disclosed that I had a malformed share in "/var/lib/samba/usershares" which prevented the checkbox from remaining checked. Thanks to loopodoopo. So the associated text in my problem description above was deleted.

UPDATE: I ran across this similar post. Network Computers Not Showing up. I applied the suggestions to "/etc/samba/smb.conf". They apparently had no effect. However, I should point out that the network does show-up but not in the form as displayed in the image "computer3.png".

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Since you have found a solution to this problem (and congratulations on describing it so clearly), please mark this thread as solved. That will make it more useful to other people who have the same problem and find your post.

Not solved yet. Look at the two attached images in my original post. The network configuration does not display correctly on computer3. The image shown for computer1 is what I believe to be the correct network configuration display and the solution that I am seeking.

What was solved; was what I had originally believed to be sympton of the problem, but it turned-out to be an independent unrelated issue. A "bad" file. Deleting that file did not fix the problem that I am experiencing.

New information.

1. I just recently installed a new router. The router is now visible in the same manner as displayed in the image "computer3.png" in my initial post. However the other computers still do not show-up.
   
2. Additionally, a bug report that I submitted on network discovery has been "confirmed" (Bug #1589975). Ubuntu 16.04 has Networking problems Whether this is part of my current problem is unknown.

Now solved.

Since installing the new router, I experimented by logging in as another user to see what the effects would be. What I found is that (as an other user) the network, popped-up quickly and appeared as shown for computer3 in the image on the first post. The obvious implication, I somehow screwed-up my (user) network configuration so that it only displays the image for computer1. That would imply that I will need to locate the personal network configuration files associated with my login.

After posting, I located a ".smb" directory in my "home" directory and deleted the ".smb" directory. On one computer it solved the problem. On the other computer, it only partially solved the problem.

For some unknown reason, the FSTAB file did play a role in not displaying the other computers on the network.

# out the line below resolved the issue of the other computers not showing-up. Why I don't know.
I also tried this approach, but it did not work either. For now, this mounting option has simply been deleted since it is not accessed that often.