
956 P.2d 98 (1998)
Deborah RUSSO, Appellant,
v.
John J. GARDNER, Respondent.
No. 29758.
Supreme Court of Nevada.
April 2, 1998.
Carol A. Menninger, Las Vegas, for Appellant.
Gus W. Flangas, Las Vegas, for Respondent.

OPINION
PER CURIAM.

FACTS
The relationship between appellant Deborah Russo ("Russo") and respondent John J. *99 Gardner ("Gardner") began sometime in 1989. Their relationship was tumultuous; Gardner moved in and out several times before 1991. During one of the times Gardner was not living with her, Russo conceived a child by another man after a "one night stand."
The child, named Zachary C. Gardner-Russo ("Zachary"), was born on May 15, 1992. At that time, Gardner was not living with Russo. However, about two weeks after Zachary was born, Gardner moved back to the home, and his relationship with Russo continued. Gardner and Russo broke up and reunited at least six times during the first three years of Zachary's life. On August 31, 1994, Russo gave birth to a second child, Samantha Russo ("Samantha"). Gardner is Samantha's biological father.
Gardner had a violent temper with Russo and Zachary. On June 19, 1995, Russo filed an application for a temporary protection order ("TPO") to keep Gardner away, alleging Gardner had physically abused her and Zachary. Russo alleged that Gardner pushed Zachary up against a wall and made him stay there. When Zachary would move, Gardner would pull down Zachary's pants and hit him. Russo also alleged Gardner physically abused her, including punching her in the stomach when she was pregnant with Samantha, and even threatened her life. Russo also stated that Gardner had been living with her for about a year, but had moved out about a week prior to June 19, 1995. On June 27, 1995, the district court entered the TPO.
Additionally, on August 6, 1995, Gardner was convicted of domestic violence against Russo.
At an unspecified time after Samantha's birth and during Gardner's cohabitation with Russo, Russo filed an application for welfare benefits. The State of Nevada filed a suit challenging paternity of Zachary and Samantha. On September 22, 1995, blood tests confirmed that Gardner was Samantha's biological father, but not the biological father of Zachary.
The relationship finally ended in December 1995. On December 18, 1995, Gardner allegedly struck Russo on the back of the head and attempted to choke her. Russo has a history of suffering from epileptic seizures, one of which was allegedly triggered by the assault. Russo was taken to the hospital, and Gardner was arrested for battery.[1]
On an unspecified date, Gardner petitioned for joint legal custody and primary physical custody of both Zachary and Samantha. On January 23, 1996, the district court entered a temporary order awarding joint legal custody to Gardner and Russo, but primary physical custody to Russo. The court also referred the case to the Family Mediation and Assessment Center ("FMAC") for evaluation.
In June 1996, during one of the times specified for a supervised exchange of the children by Russo and Gardner, Gardner physically intimidated and became verbally abusive with a court-recognized third party, Kathy Ervalina ("Ervalina"). Ervalina was transporting the children to and from their scheduled time-share with Gardner. According to Ervalina, Gardner became irate when Zachary was not present during one of the scheduled time-shares. Ervalina explained that Zachary was in surgery and could not be present. Gardner began shouting profanity at Ervalina. Ervalina told Gardner she did not have to take verbal abuse from him. Gardner then grabbed Ervalina by the arm and continued shouting profanity at her. Ervalina left the scene, went to the Las Vegas Metropolitan Police Department, and filed a complaint against Gardner for battery. On June 17, 1996, Ervalina contacted FMAC to state that she would no longer transport Zachary and Samantha to see Gardner.
On July 17, 1996, FMAC submitted a report based on its observations of Gardner, Russo, and the children. The report addressed various issues, including Gardner's allegations that Russo attempted to impede his relationship with the children and Gardner's alleged violent temper. The report indicated that there were no issues between *100 Gardner and the children that needed to be addressed. The FMAC report stated that the children were responsive to both Gardner and Russo, and that from FMAC observations, Gardner's and Russo's relationships with the children were healthy. The FMAC report also recommended Gardner seek anger management counseling and that Russo should continue counseling to work on co-parenting issues. However, the report stated that there did not appear to be any deliberate attempts by Russo to impede Gardner's relationship with the children.
On August 28, 1996, after three consecutive days of testimony in an evidentiary hearing, the district court found that Russo's testimony was motivated by animus toward Gardner and that Gardner's testimony was credible. The district court found that evidence showing Zachary called Gardner "daddy" and Russo's attempts to prevent that was further indicia of both Russo's animus and of Gardner's position as a father. The district court also found that the multiple separations the couple experienced indicated a desire to work things out and that they thought of themselves as a family. The district court concluded that Gardner had equitably and constructively adopted Zachary. Thus, the district court awarded joint legal custody of Zachary and Samantha to Gardner because Gardner had placed himself in a position of "loco parentis," and that it would be "devastating" to Zachary to have Gardner treat him differently than his sister. The district court did not cite any statutory authority in support of its findings, but instead cited to Frye v. Frye, 103 Nev. 301, 738 P.2d 505 (1987); McGlone v. McGlone, 86 Nev. 14, 464 P.2d 27 (1970); and Murphy v. Murphy, 84 Nev. 710, 447 P.2d 664 (1968). The district court also ordered that Samantha's surname be changed from "Russo" to "Gardner-Russo."
On December 18, 1996, Russo filed her notice of appeal.
Other facts have come to light since Russo appealed the district court's order. First, blood tests have confirmed that Zachary's natural father is Andrew Nati ("Nati"). Zachary and Nati have allegedly formed a "strong relationship" and enjoy spending time together. Nati has allegedly changed Zachary's birth certificate to show the child's name as "Zachary Cameron Nati," put Zachary on his health insurance coverage, notified the district attorney's office of his financial responsibility for Zachary, and changed his Social Security information to reflect his biological parentage of Zachary.

DISCUSSION
First, Russo argues in her opening brief that the district court erred in excluding testimony from a psychiatrist. However, Russo never filed any motion under NRAP 10(c) to supplement the record with any evidence to support her argument, nor did she file a reply brief in this appeal. Because Russo's argument is based solely on alleged facts outside the record, we need not consider this argument.
Next, Russo argues that the district court erred in granting visitation rights and joint legal custody of Zachary to Gardner. Specifically, Russo argues that the Nevada Uniform Parentage Act, NRS 126.011, et seq., and the parental preference presumption preclude this result and, therefore, the district court erred as a matter of law. Further, Russo contends that substantial evidence does not exist to support the district court's findings concerning the best interest of Zachary.
The district court determined that Gardner equitably and constructively adopted Zachary. Further, the written findings of fact and conclusions of law refer to both parties as "parents." Russo argues that Gardner is not a "parent" for purposes of determining custody and, therefore, the district court erred.
NRS 126.051 provides, in relevant part:
1. A man is presumed to be the natural father of a child if:
....
(b) He and the child's natural mother were cohabiting for at least 6 months before the period of conception and continued to cohabit through the period of conception.
....
(d) While the child is under the age of majority, he receives the child into his *101 home and openly holds out the child as his natural child.
....
3. A presumption under this section may be rebutted in an appropriate action only by clear and convincing evidence. If two or more presumptions arise which conflict with each other, the presumption which on the facts is founded on the weightier considerations of policy and logic controls. The presumption is rebutted by a court decree establishing paternity of the child by another man.
A trial court's determination in a child custody proceeding will not be disturbed absent an abuse of discretion, but this court must be satisfied that the trial court applied the appropriate reasoning. Litz v. Bennum, 111 Nev. 35, 37, 888 P.2d 438, 440 (1995).
We have previously considered NRS 126.051 for determining legal parentage in the context of a custody dispute between a biological parent and a non-biological parent. Hermanson v. Hermanson, 110 Nev. 1400, 887 P.2d 1241 (1994). In Hermanson, Mr. and Mrs. Hermanson were married when Mrs. Hermanson was six months pregnant. Id. at 1401, 887 P.2d at 1242. Mrs. Hermanson asserted that she told Mr. Hermanson at that time that he was not the father of the child. Mr. Hermanson denied her assertions, but also admitted Mrs. Hermanson had never actually told him that the child was his. Id. After the birth of the child, the marriage fell apart because of Mr. Hermanson's physical abuse of his wife. Id. at 1401-02, 887 P.2d at 1243. The couple divorced, and Mr. Hermanson filed a motion to be recognized as the child's "de facto" father even if he was not the biological father of the child. Id. at 1402, 887 P.2d at 1243. Subsequently, blood tests proved conclusively that Mr. Hermanson was not the biological father. Id.
However, the district court found that Mrs. Hermanson was equitably estopped from contesting the child's paternity. Id. The district court found that both Mr. and Mrs. Hermanson held themselves out to the public as the biological parents of the child, listing Mr. Hermanson as the father on the child's birth certificate and Mrs. Hermanson's welfare application. Id. at 1404-05, 887 P.2d at 1244.
We held that the fact that Mr. Hermanson admitted he was never told he was the father and his own admission in his appellate brief did not support the district court's finding. Id. at 1405, 887 P.2d at 1245. Further, "the doctrine of equitable adoption enunciated in Frye v. Frye, 103 Nev. 301, 738 P.2d 505 (1987), and the myriad of other psychological theories of parentage that the parties mention in order to determine paternity are inapplicable." Hermanson, 110 Nev. at 1406, 887 P.2d at 1245. We held that Nevada's Uniform Parentage Act was to be applied to determine legal parentage and, therefore, the district court's order granting joint legal custody must be reversed. Id. at 1406-07, 887 P.2d at 1246.
In the instant case, the district court looked to Frye and concluded that Gardner equitably adopted Zachary. The district court based this conclusion on the findings that Gardner had "placed himself in the position of `loco parentis'" and has since birth held himself out to be the biological father of Zachary.
However, as stated above, Frye was rejected by Hermanson as inapplicable for determining legal parentage in a custody proceeding. Hermanson, 110 Nev. at 1406, 887 P.2d at 1246. Further, although Gardner claims he never knew he was not the biological father, the record indicates that Gardner was never listed as Zachary's father on the birth certificate. Russo also claims she told Gardner that he was not Zachary's father at the time she was pregnant with Zachary. Russo describes the relationship as off-again, on-again, with Gardner moving in and out. Gardner asserts that he has always thought of Russo, Zachary, Samantha, and himself as a family, but admits he was not living with Russo when she told him she was pregnant with Zachary. Additionally, it is worth noting that Gardner has a documented history of physical violence with Zachary, Russo, and even an FMAC representative.
We conclude that the district court erred in applying the doctrine of equitable adoption to this case. Like Hermanson, this case *102 concerns the issue of legal custody. Frye, on the other hand, applied the equitable adoption doctrine in upholding a child support obligation. Moreover, in Frye, this court limited the application of the equitable adoption doctrine to the specific facts of that case. Frye, 103 Nev. at 301, 738 P.2d at 505.
Additionally, for purposes of determining legal parentage in a custody dispute between biological and non-biological parents, Hermanson holds that NRS 126.051 is the applicable statute. However, the district court engaged in no such analysis. Therefore, we conclude the district court committed reversible error.
Third, Russo argues that substantial evidence does not exist to support the district court's finding that Zachary's best interest is served by granting Gardner joint legal custody.
NRS 125.480 provides, in relevant part:
1. In determining custody of a minor child in an action brought under this chapter, the sole consideration of the court is the best interest of the child. If it appears to the court that joint custody would be in the best interest of the child, the court may grant custody to the parties jointly.
....
3. The court shall award custody in the following order of preference unless in a particular case the best interest of the child requires otherwise:
(a) To both parents jointly ... or to either parent....
(b) To a person or persons in whose home the child has been living and where the child has had a wholesome and stable environment.
(c) To any person related within the third degree of consanguinity ....
(d) To any other person or persons whom the court finds suitable and able to provide proper care and guidance for the child.
4. In determining the best interest of the child, the court shall consider, among other things:
(a) The wishes of the child if the child is of sufficient age and capacity to form an intelligent preference as to his custody;
...
(c) Whether either parent or any other person seeking custody has engaged in an act of domestic violence against the child, a parent of the child or any other person residing with the child.
5. Except as otherwise provided in subsection 6 or NRS 125A.360, a determination by the court after an evidentiary hearing and finding by clear and convincing evidence that either parent or any other person seeking custody has engaged in one or more acts of domestic violence against the child, a parent of the child or any other person residing with the child creates a rebuttable presumption that sole or joint custody of the child by the perpetrator of the domestic violence is not in the best interest of the child. Upon making such a determination, the court shall set forth:
(a) Findings of fact that support the determination that one or more acts of domestic violence occurred; and
(b) Findings that the custody or visitation arrangement ordered by the court adequately protects the child and the parent or other victim of domestic violence who resided with the child.
(Emphasis added.)
Here, the district court granted joint legal custody to Russo, Zachary's biological mother, and Gardner. Gardner is not Zachary's biological father, nor was he ever married to Russo. Gardner is not listed as Zachary's father on the birth certificate. Further, Gardner's relationship with Russo ended when Zachary was only three years old. The break-up was a result of continued domestic violence by Gardner against Russo and Zachary. Moreover, the break-up was preceded by Gardner moving in and out of the relationship and Russo's residence a total of six times.
Nevertheless, the district court found that it was in Zachary's best interest to be in Gardner's joint custody because Gardner had placed himself in a position of loco parentis. *103 Despite Gardner's criminal conviction for domestic violence against Russo and his history of violent behavior, the district court made no finding or consideration of any domestic violence as required by NRS 125.480(4)(c). Since Gardner was criminally convicted of domestic violence, by definition it follows that the "clear and convincing" standard was met for a finding of domestic violence under NRS 125.480(5), thereby creating a rebuttable presumption that it would not be in Zachary's best interest to be placed in Gardner's custody. The district court made no such consideration, and therefore, we conclude the district court abused its discretion.
Finally, Russo argues that the district court abused its discretion in ordering that the surname of her daughter, Samantha, be changed from "Russo" to "Gardner-Russo."
A current examination of the facts of this case reveals that Samantha is and will be living with Russo, and that Zachary's surname has changed to "Nati" since the district court's judgment. After a thorough review of the record, we conclude that there is no showing of "clear and compelling" evidence to necessitate a name change. See Magiera v. Luera, 106 Nev. 775, 777, 802 P.2d 6, 7 (1990) (holding that the "burden is on the party seeking the name change to prove, by clear and compelling evidence, that the substantial welfare of the child necessitates a name change"). Accordingly, we reverse the order changing Samantha's surname on her birth certificate. Additionally, we reverse the district court's custody order and remand this matter to the district court for proceedings consistent with the foregoing reasoning.
NOTES
[1]  Gardner does not deny he was arrested, but claims he never struck Russo. Gardner claims that Russo's brother, a Las Vegas police officer, somehow manipulated the system to have him arrested on false charges.
