Before revision:  We also consider these versions of the problem in the so-called "edge intersection model" , and obtain several interesting results. One of the resultsis an NP-hardness proof of the third version which answers a question posed by Mehrabi (WAOA 2017).
After revision:  We also consider these versions of the problem in the so-called "edge intersection model" and obtain a number of results, answering two questions posed by Mehrabi (WAOA 2017).