Completeness of the future null infinity in defining a black hole I am using these lectures by Rodnianksi and Dafermos as the reference for this question.
In third point in the list on the top of page 19 they emphasize the importance of completeness of the future null infinity in being able to define a black hole through a
Penrose diagram. I guess by completeness they mean geodesic completeness. 


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*What is the issue that is being hinted at?


In the statement just below the second diagram on page 21 they seem to be able to read off that the part of the future null infinity that is intersected by the maximal Cauchy development of their chosen Cauchy surface is incomplete.


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*I could not understand how this is obvious.


This observation is what they are using to motivate Christodoulou's thinking
of incompleteness of future null infinity as a defining criteria of naked
singularity.


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*How does this relate to the other viewpoint of Christodoulou that naked
singularity is characterized by the non-compactness of the intersection of the
past of the future null infinity with the Cauchy surface? (This paper for reference)

 A: It will be helpful if you look up the definition of a singularity in spacetime in a standard textbook on GR, Hawking/Ellis "The Large Scale Structure of Spacetime" comes to mind. 
About your first question:
The important point is that the definition of a singularity is non-trivial, it cannot simply be defined as a point on a Lorentzian manifold where some tensor diverges (despite the name "singularity", the reason for this is explained in most GR textbooks, the Hawking/Ellis book contains a particularly lucid discussion of the involved ideas). Instead, the defining property is that in the presence of a singularity, there are observers aka reference frames that do not live as long as the universe exists, they either end before the universe itself comes to an end (these are the ones who fall into the singularity) or did not exist when the universe was created (these are the ones escaping from the singularity).
The basic idea is therefore to define a spacetime without singularities to be a spacetime that is geodesically complete. Given a spacetime with a singularity, we can remove all geodesics from the spacetime that end up in the singularity and the ones that emerge from the singularity, and get a spacetime without singularities. The minimum region that we have to remove is then defined as the region comprising the singularity. Therefore the "black hole region" or "singularity" of a spacetime is defined to be the minimum subset one has to remove to get a "complete" spacetime in the sense that there are no geodesics that end or begin life prematurely.
Page 19 of your reference is a first step in making these ideas a little bit more precise, it is not a mathematical definition, but a heuristic motivation.
Penrose diagrams are a tool in this context, but not a necessary tool.
I don't understand your second question, page 21 seems to be an explanation of the above ideas using generic Penrose diagrams, this is simply a graphical explanation of what the authors have done before, the diagrams are supposed to show a spacetime with a singularity. 
As for your third question, I cannot access the Christodoulou paper.
A: At least about the definition of "complete future null infinity" I found some answers on the 8th page of these lectures by Klainerman. 
I would be glad to hear of some explanations about how the two definitions given on that page relate to each other and how are they related to the notion explained by Tim in the comments to his answer. (Tim is calling the future null infinity to be complete if the whole manifold is geodesically complete) 
Also these seem closely related to the idea of calling a hypersurface as being "generated by complete null geodesics". I would like to know what this means and why this is often used as a condition for the event horizon to satisfy. 
