Legal custody and tax custody are two different things.   The Earned Income Credit can never be claimed if the parent does not have tax custody which means that the child physically lived with them for more than half the tax year.   They can claim the child as a dependent which gives them the child exemption and child tax credit if the child is under age 17 but only if the custodial parent gives them a signed 8332 form releasing the exemption (or it is spelled out in a pre 2009 divorce decree that meets the IRS requirements).  See all of the rules below.

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Custodial Parent
These are a paraphrase of the IRS rules for divorced or separated parents that live apart.

[Note: Unless the parents have been separated at all times during the last 6 months of the year, these rules do not apply.]

See “Children of divorced or separated parents or parents who live apart” in IRS Pub 17 for full information.

https://www.irs.gov/publications/p17/ch03.html#en_US_2016_publink1000170897

This assumes that the child is under age 18 (in most states).  Once the child becomes an adult (Emancipated child), custody becomes mute and these rules no longer apply.(See examples 5 & 6 in Pub 17 for more information)

There is no such thing in the Federal tax law as 50/50, split, or joint custody. The IRS only recognizes physical custody (which parent the child lived with the greater part, but over half, of the tax year. That parent is the custodial parent; the other parent is the noncustodial parent.)

Who can claim the exemption and credits depends on who is the custodial parent. (By the IRS definition of custodial parent for tax purposes - this is not the same as the custody that a court might grant.).

The test that the IRS uses to determine the custodial parent is where the child lived for more than 1/2 (or greater part) of the year. The IRS will go so far as to require counting the nights spend in each household - that person is the custodial parent for tax purposes (if exactly equal and more than 183 days - The custodial parent is the parent with the highest AGI, if less than 183 days then neither parent has custody so the child cannot be claimed by either parent). And yes they are that picky.

See Custodial parent and noncustodial parent  under the residency test in Pub 17

https://www.irs.gov/publications/p17/ch03.html#en_US_2016_publink1000170897

Only the Custodial parent can claim: (Child would be listed as non-dependent EIC & CC only)
-Head of Household
-Earned Income Credit
-Child Care Credit

The non custodial parent can only claim: (Child would be listed as dependent)
-The Exemption
-The Child Tax Credit

But only if specifically specified in a pre-2009 divorce decree, separation agreement or the custodial spouse releases the exemption with a signed 8332 form - after 2009 the IRS only accepts a signed 8332 form that must be attached to the non-custodial parents tax return.

Note. If you are filing your return electronically, you must file Form 8332 with Form 8453, (U.S. Individual Income Tax Transmittal) for an IRS e-file Return. See Form 8453 and its instructions for more details. This must be done within 3 days of your e-filed return being accepted by the IRS.

This does NOT mean that the custodial parent can ignore any Decree or court order allowing the non-custodial parent to claim the exemption - they can be required to issue the 8332 form. They could be required by the court to do so or be in contempt.

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Post-1984 and pre-2009 divorce decree or separation agreement. If the divorce decree or separation agreement went into effect after 1984 and before 2009, the noncustodial parent may be able to attach certain pages from the decree or agreement instead of Form 8332. The decree or agreement must state all three of the following.

1. The noncustodial parent can claim the child as a dependent without regard to any condition, such as payment of support.
2. The custodial parent will not claim the child as a dependent for the year.
3.The years for which the noncustodial parent, rather than the custodial parent, can claim the child as a dependent.

The noncustodial parent must attach all of the following pages of the decree or agreement to his or her tax return.

* The cover page (write the other parent's social security number on this page).
* The pages that include all of the information identified in items (1) through (3) above.
* The signature page with the other parent's signature and the date of the agreement.

Post-2008 divorce decree or separation agreement.   The noncustodial parent cannot attach pages from the decree or agreement instead of Form 8332 if the decree or agreement went into effect after 2008. The custodial parent must sign either Form 8332 or a similar statement whose only purpose is to release the custodial parent's claim to an exemption for a child, and the noncustodial parent must attach a copy to his or her return. The form or statement must release the custodial parent's claim to the child without any conditions. For example, the release must not depend on the noncustodial parent paying support.

Form 8332 rules:
Attach this form or similar statement to your tax return for each year you claim the exemption for your child. You can claim the exemption only if the other dependency tests in your tax return instruction booklet are met.

Note. If you are filing your return electronically, you must file Form 8332 with Form 8453, (U.S. Individual Income Tax Transmittal for an IRS e-file Return). See Form 8453 and its instructions for more details.

This must be done within 3 days of your e-filed return being accepted by the IRS.
http://www.irs.gov/pub/irs-pdf/f8453.pdf