When there is no signed document by the "custodial parent"---then the IRS recognizes the custodial parents claim to dependency. However, most parents state that they have "joint custody" or "50/50 custody". In reality, usually one parent has the child(ren) longer that the other, especially since each year has 365 days, (except for last year---366). The IRS recognizes overnights as a tool for counting time spent with each parent.
If there is a true situation that each parent had the child for exactly 183 days during the year, and the parents were separate for the entire year, then the child would not be considered a "qualifying child" of either parent, and could only be a qualified relative. In this case, whoever paid more than 50% of the child's support would be able to claim the child.
If the situation is that the parents were together for part of the year, and the child alternated between them for part of the year split up, and was with each parent exactly the same # XXXXX days, (For example 240 days with one parent, 240 days with the other parent---because of shared time for part of the year) then there is a tie-breaker situation and whoever had the highest adjusted gross income would be allowed to claim the child.
There is a very convoluted set of rules for this type of situation. You can read m ore about types of dependency, rules for children of divorced or separated parents, and tie-breaker situations in Publication 501, found at www.irs.gov. If you need additional clarification, please let me know.