New Approach

I've been trying to change the way I program, and this is the first real attempt I've made.  It's a simple function - deleting an edge. However, instead of just diving in and coding and running and see what works and what errors I get, I sat down and wrote out on paper what I'd have to do. I took into account as many possible boundary cases as I could think of and sketched exactly what I had to do. Here's the final version of what I wrote out:

deleteEdge(edge, edgeContainer)  

    1) Remove edge from edgeContainer (SVG object) - it can't be seen now.
    2) Extract row edge.idx from the database, getting its Source, Destination, and Title. [destination, source, and title are column names of my database]
    3) Attempt to extract similarEntry from the database, where destination = Destination (from step 2) and source != Source (from step 2). This is checking for if there is another edge that leads to Destination.
  
    if (similarEntry is not NULL) {
        // Then the Destination of this edge to be deleted has at least one other edge leading to it, so after deletion it will NOT become a root node.
        if (similarEntry's row > edge.idx [it's row number]) {
            // This means that the edge to be deleted is the first entry in the database where Destination (from step 2) is in the destination column. This is important because the user altered title for each website is stored in that websites first occurance in the database. So, we have to copy the title from this entry to the next reference of Destination - similarEntry (since we know it comes after it since its row number is greater).
            4) Update row similarEntry to have title = Title (from step 2).
        }
        5) Delete row edge.idx from the database.
    } else { 
        // There is NO other entry in the database where destination = Destination and source != Source, so the Destination of this edge we're deleting will become a root node.
       6) Update row edge.idx so that source='NULL', making it a root. 
    }
   
    7) Reload the graph_page.html, to reset the forces.
    8) Done.

    Once I had this all written up writing the code was simple, and aside from a few typos it worked perfectly on the first run. This is a major improvement over my standard method of writing the first idea that comes in to my head, then running it and fixing bugs over and over until it works.