Possibly the single most difficult task a head judge will need to deal with is the Disqualification. It's so rarely needed that many judges I know haven't even needed to DQ a player in their career. Yet when the time comes to do so, a judge needs to be prepared to act decisively.
The first step to being prepared is to know where you draw the line. The DCI has set out guidelines, but each judge has a slightly different idea of what constitutes severe unsportsmanlike conduct or stalling, the two most likely scenarios with a DQ penalty. If you as a judge work hard to control the lesser versions, minor unsportsmanlike conduct and slow play, you can further reduce the probability of a DQ. Other scenarios are easier to identify, such as cheating by adding cards to a sealed deck, or overhearing a bribery attempt.
If the possible DQ was brought to your attention by another judge the first thing to do is get the full story from that judge, then confirm the facts. During your investigation it's important to assign another experienced judge to cover for you while you are occupied. It's also important to prevent the typical "judge cluster" that gathers during an interesting call. Letting a judge in advanced training observe can be a good idea, but letting every judge at the event listen in is an invitation to disaster. I sometimes also let the TO know about the impending DQ, in case removal from the site might be needed.
During your investigation it's vital to keep an open mind. Approaching a player by saying "I'm going to DQ you, do you have anything to say?" is certainly not good. It's much better to approach the player in question and ask them to explain their situation, and then present your ruling. By calmly listening to the explanation and carefully considering your actions the player being DQ'ed will have a much easier time remaining calm.
Once you've identified that a DQ is appropriate it's time to take action. My first step is to separate the DQ'ed player from the crowd. If the facts surrounding the DQ don't involve the other player (such as adding cards, etc) it may be better to separate the player as soon as you learn of the offense. Knowing a calm, quiet, secluded place to talk to the DQ'ed player can help keep the situation calm. Many players that would be loud and obnoxious in public will be much more reasonable when they don't feel the need to maintain their public facade.
The next step is to carefully explain to the player your view of their action, and why you must disqualify them. No matter what you do some players will continue to argue with you, and the best way to deal with them is to quietly inform them they may contact the DCI, but for now your decision is final. If they continue to argue it's usually best to let them know they are no longer welcome on site and show them to the nearest exit, again remaining calm cool and collected at all times.
Most players will react much better than that, and will usually admit their wrongdoing and apologize. It's important not to waiver in your determination to DQ, but certainly acknowledge that their actions post DQ were appropriate. If a judge backs down on a DQ (without a change in facts of the case) then the DQ becomes a watered down penalty. Since there is no more significant penalty an individual judge can hand out, it's vital to keep the DQ as a dreadful penalty. Don't wield it like a club, but a player should understand that they have done a significant wrong when a DQ is handed out, and it has real meaning with you as a judge.
After you have finished dealing with the player the next step is to collect all the notes and material you based your decision on and write an incident report. In addition to the brief description of the DQ in DCI Reporter (assuming you are using it) you should write a much more detailed description of the facts so that it can be reported to the DCI. The DCI may determine that a further investigation is needed, and your report is the first step to making that decision. Without an unbiased, detailed description of the DQ written down as soon as possible it is likely that details will be forgotten. Once you have written up your description you need to send it to the DCI. Someone will contact you if more information is needed, however a good incident report should rarely need follow up from the DCI. Your incident report should be emailed to John Grant at email@example.com or mailed in if you have no email access.
No matter how much you run through a DQ scenario in your head, nothing beats actual experience. The next time you have the chance to work with a more experienced judge be sure to ask them about how they deal with difficult situations, and let them know you are interested in gaining experience. Experience with DQing a player is vital if you want to advance to high levels of certification.