This would be dependent upon the amount of invalid s'chach, if it is more than four tefachim or less (Shulchan Aruch, section 1).
Can you explain the difference?
If the area of invalid s'chach is four by four tefachim it would be prohibited to sit underneath it. Furthermore, there are times when the entire succah can be invalidated. For example, when we are dealing with a succah that has three walls, and the invalid s'chach is four tefachim wide and extends all the way from the middle wall until the end of the succah (see diagram), the succah would be prohibited. The reason for this is because we say that this area of invalid s'chach divides the succah into two separate parts, giving us two individual succahs with each one having only two walls. Such a succah would be prohibited (M.B., section 2).3. What would the halachah be if the invalid s'chach was less than four tefachim?
However, if the invalid s'chach is situated beyond the area which constitutes a kosher succah, for instance, if the four tefach wide strip of invalid s'chach extends from wall to wall parallel to the middle wall, and more than seven tefachim away from it, then it would be permitted to use that area.
If the invalid s'chach was between three and four tefachim the succah would be kosher (but see next question in reference to a small succah that has only the minimum area of seven by seven tefachim). However, there is a disagreement as to the permissibility of sleeping under this area; therefore it is preferable to be stringent and not use that area. If the area of invalid s'chach is less than three tefachim it is certainly permitted to sleep under that area (M.B. section 3). (From the Chazon Ish it would appear that it would be permitted to sleep under invalid s'chach even when it is between three and four tefachim. (See Arugas Habosem 6.)4. Do the above halachos apply to all succos?
No; when the succah has the minimum size of seven by seven tefachim the halachah would be different. In such a case s'chach measuring three tefachim would prohibit the entire succah. This halachah would apply regardless of whether the invalid s'chach is located in the middle or along the sides of the succah.8. Why is it that invalid s'chach which is less than four amos is not considered as separating between the succah wall and the valid s'chach, whereas we said before in question 2 that invalid s'chach located in the middle of the succah separates the succah into two parts when it is just more than four tefachim?
The Mishnah Berurah in the name of the Magen Avraham adds (section 8), that this stringency can apply to a succah which is larger than seven by seven as well, if we subtract the area of invalid s'chach we are not left with the minimum measurement of a kosher succah (seven by seven tefachim) the succah would be prohibited. As a result of this halachah even if a succah has a size of up to ten tefachim it could still be prohibited by a strip of s'chach three tefachim wide. The reason is that after subtracting the invalid s'chach you still are not left with a succah containing seven by seven tefachim.
The Chazon Ish, however, suggests that in a succah which is between seven and ten tefachim in size, if the amount of invalid s'chach is less than four tefachim (which, if it would be on a large succah, it would be permitted to sleep underneath it), we could then say that as long as not more than an area of three tefachim (of the minimum of seven tefachim) is missing from the small succah we would say that the invalid s'chach can complete the minimum measurement of seven by seven for the succah. The reason for this is because we can not give the invalid s'chach more validity when it is located on a large succah then when it is located upon a small one. The Chazon Ish, however, remains doubtful concerning this halachah.
The difference is as follows. When the invalid s'chach is located along the sides of the succah and is less than four amos we can apply the principle of "the bent wall" (dofen amukah). This principle allows us to consider the wall as if it is bent over and extends up to the valid s'chach; thus the invalid s'chach is considered as part of the wall itself. This principle is a halachah LeMoshe MiSinai (Shulchan Aruch, section 1) (see diagram).9. In the previous question, would it be permitted to sit under the invalid s'chach which we said becomes part of the wall through the principal of "the bent wall" (dofen amukah)?
Therefore, a house whose roof was removed in the middle and s'chach was placed there, as long as the remaining roof area (which is invalid s'chach) surrounding the s'chach is less than four amos between the s'chach and the walls of the house the succah is kosher (Shulchan Aruch, section 1). (See next question.)
There is a difference here between a large and small succah. In reference to a large succah (meaning that besides the invalid s'chach there is at least an area of seven by seven of valid s'chach), we would say that an area of at least four by four tefachim of invalid s'chach is considered an important entity unto itself and can not become nullified to the rest of the succah. As a result of this it would be prohibited to sit under the invalid s'chach even though it is considered part of the wall.Next Chapter
However, it should be noted that in reference to a small succah (meaning one whose entire area is only seven by seven tefachim) we say that the invalid s'chach, if it measures an area of three by three tefachim, invalidates the succah, no matter if it located on the side of the succah or in the middle. (For additional information regarding a small succah see above, question 4.)