
755 N.W.2d 376 (2008)
276 Neb. 520
The COUNTY OF SARPY, Nebraska, appellant,
v.
The CITY OF GRETNA, Nebraska, appellee.
No. S-07-498.
Supreme Court of Nebraska.
September 5, 2008.
*378 Lee K. Polikov, Sarpy County Attorney, and Michael A. Smith for appellant.
John K. Green, Omaha, for appellee.
HEAVICAN, C.J., WRIGHT, CONNOLLY, GERRARD, STEPHAN, McCORMACK, and MILLER-LERMAN, JJ.
MILLER-LERMAN, J.

NATURE OF CASE
The City of Gretna sits entirely within the borders of Sarpy County, Nebraska. This challenge by Sarpy County, related to annexation ordinances enacted by Gretna, is before us for the third time. In County of Sarpy v. City of Gretna, 267 Neb. 943, 678 N.W.2d 740 (2004) (Sarpy I), we concluded that Sarpy County had standing to challenge the annexations. We reversed the judgment of the district court for Sarpy County which had dismissed the action, and we remanded the cause for further proceedings. On remand, the district court found that the annexation ordinances were valid and further found that, in any event, Sarpy County had not produced evidence of damages.
*379 Sarpy County appealed and claimed as its sole assignment of error that the district court had erred in concluding that Gretna's annexation ordinances were valid. On appeal, we concluded that the annexation ordinances were invalid because the lands Gretna sought to annex were not contiguous or adjacent to the corporate limits of Gretna. See County of Sarpy v. City of Gretna, 273 Neb. 92, 727 N.W.2d 690 (2007) (Sarpy II). We reversed the judgment of the district court and remanded the cause with directions to enter judgment consistent with our opinion.
On remand, Sarpy County filed a motion for accounting, seeking an order for an accounting for fees collected by Gretna. An accounting had been requested by Sarpy County as relief in its amended petition in Sarpy I. On April 4, 2007, the district court entered judgment on our mandate without addressing Sarpy County's motion for accounting.
Sarpy County appeals and asserts that the district court erred in failing to address its motion for accounting. We conclude that issues with regard to an accounting were waived by Sarpy County when it did not raise those issues on appeal in Sarpy II; therefore, such issues were not part of our mandate and the district court did not err when it did not address such issues on remand. We affirm.

STATEMENT OF FACTS
The facts of the underlying dispute are set forth more fully in Sarpy I and Sarpy II. In sum, the Gretna City Council adopted ordinances by which it sought to annex certain lands in Sarpy County. The lands Gretna sought to annex included strips of certain highways. On June 20, 2002, Sarpy County filed a petition in the district court challenging the annexations on various bases. Sarpy County filed an amended petition on August 12 in which it prayed for relief including, inter alia, an order for an accounting of sums collected by Gretna since July 3, 2001, for rezoning applications, building permit fees, and other zoning fees for the areas purportedly annexed pursuant to the ordinances. Gretna filed a demurrer to the amended petition asserting various bases. The district court sustained the demurrer on the basis that Sarpy County lacked standing to bring the action. Sarpy County appealed. We concluded that Sarpy County had standing to challenge the annexations. We reversed the judgment of the district court and remanded the cause for further proceedings. See Sarpy I.
On remand, the district court conducted a bench trial. Evidence was adduced regarding the annexations and the nature of the land encompassed thereby. A planning and zoning administrator for Sarpy County also testified as to various fees Sarpy County would have collected but for the annexations. In its judgment entered May 23, 2005, the district court found that the ordinances were valid and further stated, "Even though the ruling was in favor of the Defendant [Gretna], the Plaintiff [Sarpy County] did not produce evidence of damages of any specific losses." The district court entered judgment against Sarpy County and dismissed the action. Sarpy County appealed. In Sarpy II, Sarpy County's sole assignment of error on appeal was that the district court erred in entering judgment in favor of Gretna "where the undisputed evidence clearly showed that the statutory requirements of adjacency and contiguity of lands to be annexed to a second-class city were not met." We stated that "[t]he single issue presented in this appeal is whether the two parcels of land which Gretna sought to annex were contiguous or adjacent to its existing corporate limits." 273 Neb. at 95, *380 727 N.W.2d at 694. We concluded that the annexations were invalid and void because they did not meet the contiguity or adjacency requirement of Neb.Rev.Stat. § 17-405.01 (Reissue 1997). We therefore reversed the judgment of the district court and remanded the cause "with directions to enter judgment consistent with this opinion." 273 Neb. at 98, 727 N.W.2d at 696.
Our decision in Sarpy II was filed February 23, 2007. Our mandate was filed in the district court on April 2. That same day, Sarpy County filed a motion for accounting. In its motion, Sarpy County noted that its amended petition asked for an accounting as relief sought and asserted that the district court's prior order did not order an accounting because the court did not find Gretna's ordinances invalid and did not reach the issue whether Sarpy County was entitled to an accounting. Sarpy County asserted that an accounting was appropriate given the evidence adduced in the case and the mandate filed therein. Sarpy County requested that the court order an accounting. On April 4, the district court entered a "Judgment on Mandate" in which it entered judgment pursuant to this court's opinion in Sarpy II which related solely to the propriety of the annexations. The district court did not address Sarpy County's motion for accounting in the April 4 judgment or elsewhere.
Sarpy County appeals the April 4, 2007, order.

ASSIGNMENT OF ERROR
Sarpy County asserts that the district court erred when it failed to fully address Sarpy County's motion for accounting and related issues prior to issuance of the final judgment on mandate.

STANDARDS OF REVIEW
The construction of a mandate issued by an appellate court presents a question of law. Pennfield Oil Co. v. Winstrom, 276 Neb. 123, 752 N.W.2d 588 (2008). On questions of law, we are obligated to reach a conclusion independent of the determination reached by the court below. Id.

ANALYSIS
Sarpy County contends that the district court should have considered its motion for accounting on remand pursuant to our mandate in Sarpy II. We conclude that the district court ruled against Sarpy County on the accounting issue in the order appealed from in Sarpy II, that Sarpy County failed to assign error to such ruling, and that our mandate in Sarpy II was not broad enough for the district court to permit Sarpy County to relitigate issues it had waived on appeal. Accordingly, we affirm.
The primary legal issue in this appeal is whether a consideration of Sarpy County's motion for accounting was within the scope of our remand to the district court in Sarpy II. After receiving a mandate, a trial court is without power to affect rights and duties outside the scope of the remand from an appellate court. Pennfield Oil Co., supra. In Sarpy II, we remanded the cause to the district court "with directions to enter judgment consistent with this opinion." 273 Neb. at 98, 727 N.W.2d at 696. When an appellate court's mandate makes its opinion a part thereof by reference, the lower court should examine the opinion with the mandate. This allows the lower court to determine the judgment to be entered or the action to be taken thereon. Pennfield Oil Co., supra. Thus, we examine our opinion in Sarpy II to determine whether our mandate permitted the district court to consider *381 Sarpy County's motion for accounting on remand.
In our opinion in Sarpy II, we noted that "[t]he single issue presented in this appeal is whether the two parcels of land which Gretna sought to annex were contiguous or adjacent to its existing corporate limits." 273 Neb. at 95, 727 N.W.2d at 694. We concluded that the annexations were invalid and void because they did not meet the contiguity or adjacency requirement of § 17-405.01. We remanded with directions to enter judgment consistent with the opinion; because the opinion addressed only the issue of whether the annexations were valid, the only judgment to be entered by the district court consistent with the opinion was an order declaring the annexations invalid and void.
Sarpy County argues that its request for an accounting should have been considered by the district court on remand, because it was still an open issue or the issue was reopened when we determined that the annexations were invalid and void. However, we note that in the May 23, 2005, order appealed from in Sarpy II, the district court stated that although it found in favor of Gretna with regard to the validity of the annexations, it nevertheless further found that Sarpy County "did not produce evidence of damages of any specific losses." We read this portion of the order as a ruling on the issue of an accounting in which the court found that Sarpy County failed, despite certain evidence, to prove entitlement to such relief. Sarpy County did not assign error to this ruling in its appeal in Sarpy II, and we therefore determine that such ruling became the law of the case with regard to an accounting.
The law-of-the-case doctrine reflects the principle that an issue that has been litigated and decided in one stage of a case should not be relitigated at a later stage. Pennfield Oil Co. v. Winstrom, 276 Neb. 123, 752 N.W.2d 588 (2008). The doctrine promotes judicial efficiency and protects the parties' settled expectations by preventing parties from relitigating settled issues within a single action. Id. The doctrine applies with greatest force when an appellate court remands a case to an inferior tribunal. Id. Upon remand, a district court may not render a judgment or take action apart from that which the appellate court's mandate directs or permits. Id.
We have recognized that under the mandate branch of the law-of-the-case doctrine, a decision made at a previous stage of litigation, which could have been challenged in the ensuing appeal but was not, becomes the law of the case; the parties are deemed to have waived the right to challenge that decision. Id. An issue is not considered waived if a party did not have both an opportunity and an incentive to raise it in a previous appeal. Id. Also, we have recognized that an exception to the law-of-the case doctrine applies if a party shows a material and substantial difference in the facts on a matter previously addressed by an appellate court. Id.
We conclude that Sarpy County waived the issue of an accounting when it failed to assign error to the district court's ruling against it on the damage issue in Sarpy II. Sarpy County had the opportunity to raise the issue on appeal in Sarpy II and had incentive to do so in order to preserve the issue in the event this court ruled in its favor on the issue of the validity of the annexations. There is no credible assertion that there has been a material and substantial difference in the underlying facts justifying an exception to the law-of-the-case doctrine. Sarpy County argues that there is a difference now because the annexations have been declared invalid and an accounting has become an issue of *382 greater relevance than at the time of the May 23, 2005, order. We find this argument unpersuasive. In Sarpy II, Sarpy County appealed from the district court's May 23 order in which the court had determined that the annexations were valid and that Sarpy County had proved no damages. At the time of the appeal in Sarpy II, it was apparent that the issue of entitlement to an accounting and damages would become relevant if this court held in Sarpy County's favor on the annexation issue. Sarpy County should have assigned error to the district court's finding of no damages in order to preserve the issue for further proceedings; Sarpy County did not assign such error. Because Sarpy County waived the damages issue on appeal in Sarpy II, the district court's finding of no damages in the May 23 order stands as the law of the case.
An appellant waives claims that were decided against it by the trial court if the appellant elects not to raise those issues on appeal. Pennfield Oil Co., supra. Sarpy County waived the accounting issue by failing to assign error to the district court's finding of no damages in the May 23, 2005, order from which it appealed in Sarpy II. The district court's finding on damages in the May 23 order stands as the law of the case. The issue was not part of our mandate on remand, and the district court did not err when it did not address Sarpy County's motion for accounting.

CONCLUSION
Sarpy County waived its challenge to the district court's finding of no damages when it failed to assign error to the finding in its appeal in Sarpy II. Our mandate in Sarpy II was not broad enough to permit Sarpy County on remand to relitigate the law of the case regarding damages. We therefore conclude that the district court did not err when it did not address Sarpy County's motion for accounting, and we affirm the district court's order entering judgment in accordance with our mandate in Sarpy II.
AFFIRMED.
