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I have a vector of lines, in which some of them are isolated from the rest of the lines, but I need that every line is connected to at least another one, so as to obtain a unique connected path.

This is a part of the network, where there are isolated (black) lines:Disconnected_network

I would like to know if there is a tool that could connect automatically those lines in order to obtain a unique path (for example, as I shown with the evident handmade red lines)

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You could try using the Join multiple lines plugin which allows you to select multiple lines and it will automatically create a new line from the existing vertices. You can install this from the menubar:

Plugins > Manage and Install Plugins

Example:

Select your disconnected lines (highlighted in yellow):

Disconnected lines

Then click the icon for the Join multiple lines plugin:

New connected line

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