Dynamic Graphics Object Painting

There are various strategies you might pursue for this.

1. If the objects are never removed from the drawing once done, use a BufferedImage, put it in a (ImageIcon in a) JLabel. When it comes time to update:
   1. Get the graphics instance of the image and draw the new element.
   2. Dispose of the graphics object.
   3. Call repaint() on the label.
2. Keep a list of the drawn elements. In the paint method, paint them all. When a new element is added, call repaint() on the rendering component.

Here is an example of the 1st technique:

import java.awt.image.BufferedImage;
import java.awt.*;
import java.awt.event.*;
import javax.swing.*;
import java.util.Random;

public class MyCanvas
{
    JLabel view;
    BufferedImage surface;
    Random random = new Random();

    public MyCanvas()
    {
        surface = new BufferedImage(600,400,BufferedImage.TYPE_INT_RGB);
        view = new JLabel(new ImageIcon(surface));
        Graphics g = surface.getGraphics();
        g.setColor(Color.ORANGE);
        g.fillRect(0,0,600,400);
        g.setColor(Color.BLACK);
        // Keep this until I figured out if it's painted on load or not.
        g.drawLine(10, 20, 350, 380);
        g.dispose();

        ActionListener listener = new ActionListener() {
            public void actionPerformed(ActionEvent ae) {
                addNewElement();
            }
        };
        Timer timer = new Timer(200, listener);
        timer.start();
    }

    public void addNewElement() {
        boolean drawArc = random.nextBoolean();
        int x = random.nextInt(60);
        int y = random.nextInt(40);
        Graphics g = surface.getGraphics();
        if (drawArc) {
            g.setColor(Color.BLUE);
            int xx = random.nextInt(60);
            int yy = random.nextInt(40);
            drawArc(x,y,xx,yy,g);
        } else {
            drawNode(x,y,g);
        }
        g.dispose();
        view.repaint();
    }

    public static void main(String[] args)
    {
        MyCanvas canvas = new MyCanvas();
        JFrame frame = new JFrame();
        int vertexes = 0;
        // Change this next part later to be dynamic.
        vertexes = 10;
        int canvasSize = vertexes * vertexes;
        frame.setSize(canvasSize, canvasSize);
        frame.setDefaultCloseOperation(JFrame.EXIT_ON_CLOSE);
        frame.setContentPane(canvas.view);
        frame.pack();
        frame.setLocationByPlatform(true);
        frame.setVisible(true);
    }

    public void drawNode(int x, int y, Graphics g)
    {
            // Treat each location as a 10x10 block. If position 1,1 then go to (5,5) - If position 3,5 then go to (25, 45) eg: (x*10)-5, (y*10)-5
            int xLoc = (x*10) - 5;
            int yLoc = (y*10) - 5;
            g.setColor(Color.white);
            g.fillOval(xLoc, yLoc, 8, 8);
            g.drawOval(xLoc, yLoc, 8, 8);
    }

    public void drawArc(int x, int y, int xx, int yy, Graphics g)
    {
            int xLoc = (x*10) - 5;
            int yLoc = (y*10) - 5;
            int xxLoc = (xx*10) - 5;
            int yyLoc = (yy*10) - 5;
            g.drawLine(xLoc, yLoc, xxLoc, yyLoc);
    }
}

Further tip

You might notice that the lines look quite ‘jagged’ & ugly. Both the BufferedImage or a JComponent has access to the more useful Graphics2D object (for the JComponent it is necessary to cast it in paintComponent()). A Graphics2D instance accepts rendering hints that can be used to smooth (dither) the elements drawn.