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A very thin cover: Hats that are woven of straw [and hence are very light and transparent] are considered a valid head covering.
Is there a minimum size for a Yarmulke? How much of the head must be covered?
Some Poskim rule that the Yarmulke must cover majority of the circumference of the Karkafta [i.e. skull] of the head. Other Poskim rule that the Yarmulke does not need to be this large, and rather as long as it is apparent to others from all sides of the head, it is valid. However during prayer some write that according to all, one must cover majority of the head. According to all those Yarmulkes which are very small and are not recognizable to others is invalid.
If one is wearing a toupee must he wear a Yarmulke?
Is a transparent or camouflage head covering valid?
If the head covering is very difficult to recognize then it has the same status as a toupee of which there is a dispute amongst Poskim. Nevertheless in a time of need one can be lenient after consultation with a Rav.
Is a baseball cap and hats of the like a valid covering?
Yes. [However it is best to wear a Yarmulke in the open when possible.]
Is a netted Yarmulke valid?
If the Yarmulke is recognizable then it is valid even if it is perforated and contains more empty space than actual material.
 Halef Lecha Shlomo 3; Divreiy Yatziv 1/60; Birchas Habayis 1/22; Likkutei Mahrich Seder Levishas Begadim; Piskeiy Teshuvos 2/9
 Igros Moshe 1/10; See also Yechaveh Daas 4/1; Tzitz Eliezer 13/13
 So explains Piskeiy Teshuvos 2/9
 Yechaveh Daas 4/1; Tzitz Eliezer 13/13
 Piskeiy Teshuvos ibid; Divrei Yisrael 1/18
 Kaf Hachaim 2/20; Artzos Hachaim brought in M”B 2/12
Story of Rebbe Maharash: See Reshimos Devarim 3 p. 146-149 that the Rebbe Maharash once had to enter a medical bath resort which had a mandatory uncovered head policy. The Rebbe took off his Yarmulke and entered into the resort with an uncovered head. A certain Gadol within the Mussar world was also present at that time and asked the Rebbe Maharash how he could be so haste to remove his head covering. The Rebbe Maharash then showed him that he is wearing a wig and under the wig is a Yarmulke. Based on this story it ends up that the Rebbe Maharash held that in a time of need it is permitted to wear an undisguised head covering and we do not suspect for Maaras Ayin. Nevertheless the Rebbe was still particular to wear a Yarmulke under the wig and did not suffice with the wig as a head covering. Whatever the case, one should not deduce Halachic matters from an oral story as many times the story tellers were not particular of all the details involved as well as that stories traditionally evolve throughout the generations.
 Peri Megadim; M”B 2/12; Sdei Chemed Klalim Mareches 3; Toras Chaim Sofer 5; Piskeiy Teshuvos 2 footnote 57
 M”B 2/12
 Piskeiy Teshuvos 2 footnote 57
 Admur 91/4; Igros Moshe 4/2
 As one of the reasons mentioned above is so one appear like a Jew and when one wears a common hat this is not recognizable.
 Chasam Sofer 6/2 brought in Sdei Chemed Brachos 1/42; Piskeiy Teshuvos 2 footnote 57