Question: To satisfy what dynasty, did the Duchy of Pomerania get partitioned several times who was later granted as a whole and the Principality of Rügen as a fief to the dukes of both Pomerania-Stettin and Pomerania-Wolgast, by a King of Bohemia and the first King of Bohemia to also become Holy Roman Emperor?

Context paragraphs:
German–Polish War (1002–18): The German–Polish War which took place from 1002 to 1018 consisted of a series of struggles between the Ottonian king Henry II of Germany (Holy Roman Emperor from 1014) and the Polish Piast ruler Bolesław I the Brave.  The locus of conflict was the control of Lusatia, Upper Lusatia, as well as Bohemia, Moravia and Slovakia.  The fighting ended with the Peace of Bautzen in 1018, which left Lusatia and Upper Lusatia as a fief to Poland, and Bohemia became a duchy in the Holy Roman Empire.

Treaty of Stettin (1630): The Treaty of Stettin (Swedish: "Traktaten" or "Fördraget i Stettin") or Alliance of Stettin (German: "Stettiner Allianz" ) was the legal framework for the occupation of the Duchy of Pomerania by the Swedish Empire during the Thirty Years' War.  Concluded on 25 August (O.S.) or 4 September 1630 (N.S.), it was predated to 10 July (O.S.) or 20 July 1630 (N.S.), the date of the Swedish Landing.  Sweden assumed military control, and used the Pomeranian bridgehead for campaigns into Central and Southern Germany.  After the death of the last Pomeranian duke in 1637, forces of the Holy Roman Empire invaded Pomerania to enforce Brandenburg's claims on succession, but they were defeated by Sweden in the ensuing battles.  Some of the Pomeranian nobility had changed sides and supported Brandenburg.  By the end of the war, the treaty was superseded by the Peace of Westphalia (1648) and the subsequent Treaty of Stettin (1653), when Pomerania was partitioned into a western, Swedish part (Western Pomerania, thenceforth Swedish Pomerania), and an eastern, Brandenburgian part (Farther Pomerania, thenceforth the Brandenburg-Prussian Province of Pomerania).

Capitulation of Franzburg: The Capitulation of Franzburg (German: "Franzburger Kapitulation" ) was a treaty providing for the capitulation of the Duchy of Pomerania to the forces of the Holy Roman Empire during the Thirty Years' War.  It was signed on 10 November (O.S.) or 20 November (N.S.) 1627 by Bogislaw XIV, Duke of Pomerania and Hans Georg von Arnim, commander in chief of an occupation force belonging to the army of Ferdinand II, Holy Roman Emperor, led by Albrecht von Wallenstein.  While the terms of the capitulation were unfavourable for the Duchy of Pomerania already, occupation became even more burdensome when the occupation force did not adhere to the restrictions outlined in Franzburg.  Stralsund resisted with Danish, Swedish and Scottish support, another Danish intervention failed.  Imperial occupation lasted until Swedish forces invaded in 1630, and subsequently cleared all of the Duchy of Pomerania of imperial forces until 1631.

Pomerania-Stolp: Pomerania-Stolp (German: "(Teil-)Herzogtum Pommern-Stolp" , Polish: "księstwo słupskie, "Duchy of Słupsk"" ) was one of the partitions of the Duchy of Pomerania (German: "Herzogtum Pommern" ).  Centered in Słupsk, it was created from another partition of the Duchy of Pomerania, Pomerania-Wolgast, to satisfy Bogislaw V, Duke of Pomerania in 1368, and existed until 1459, when it was inherited by Eric II of Pomerania-Wolgast.  In 1474, it was merged to the partition of Bogislaw X, Duke of Pomerania, who four years later became the sole duke of Pomerania.

Charles IV, Holy Roman Emperor: Charles IV (Czech: "Karel IV."  , German: "Karl IV."  , Latin: "Carolus IV" ; 14 May 1316 – 29 November 1378), born Wenceslaus, was a King of Bohemia and the first King of Bohemia to also become Holy Roman Emperor.  He was a member of the House of Luxembourg from his father's side and the House of Přemyslid from his mother's side, which he emphasised, because it gave him two saints as direct ancestors.

Principality of Rügen: The Principality of Rügen (German: "Fürstentum Rügen" ) was a Danish principality consisting of the island of Rügen and the adjacent mainland from 1168 until 1325.  It was governed by a local dynasty of princes of the "Wizlawiden" ("House of Wizlaw") dynasty.  For at least part of this period, Rügen was subject to the Holy Roman Empire.

Margaret of Bohemia, Burgravine of Nuremberg: Margaret of Bohemia (29 September 1373 – 4 June 1410) was the younger daughter of Holy Roman Emperor Charles IV and his fourth wife Elizabeth of Pomerania.  Her siblings included Anne of Bohemia and Holy Roman Emperor Sigismund.

List of monarchs of Prussia: The monarchs of Prussia were members of the House of Hohenzollern who were the hereditary rulers of the former German state of Prussia from its founding in 1525 as the Duchy of Prussia.  The Duchy had evolved out of the Teutonic Order, a Roman Catholic crusader state and theocracy located along the eastern coast of the Baltic Sea.  The Teutonic Knights were under the leadership of a Grand Master, the last of whom, Albert, converted to Protestantism and secularized the lands, which then became the Duchy of Prussia.  The Duchy was initially a vassal of the Kingdom of Poland, as a result of the terms of the Prussian Homage whereby Albert was granted the Duchy as part of the terms of peace following the Prussian War.  When the main line of Prussian Hohenzollerns died out in 1618, the Duchy passed to a different branch of the family, who also reigned as Electors of Brandenburg in the Holy Roman Empire.  While still nominally two different territories, Prussia under the suzerainty of Poland and Brandenburg under the suzerainty of the Holy Roman Empire, the two states are known together historiographically as Brandenburg-Prussia.  Following the Second Northern War, a series of treaties freed the Duchy of Prussia from any vassalage to any other state, making it a fully sovereign Duchy in its own right.  This complex situation (where the Hohenzollern ruler of the independent Duchy of Prussia was also a subject of the Holy Roman Emperor as Elector of Brandenburg) laid the eventual groundwork for the establishment of the Kingdom of Prussia in 1701.  For diplomatic reasons, the rulers of the state were known as the King in Prussia from 1701 to 1772; largely because they still owed fealty to the Emperor as Electors of Brandenburg, the "King in Prussia" title (as opposed to "King of Prussia") avoided offending the Emperor.  As the Prussian state grew through several wars and diplomatic moves throughout the 18th century, it became apparent that Prussia had become a Great Power that did not need to submit meekly to the Holy Roman Empire.  By 1772, the pretense was dropped, and the style "King of Prussia" was adopted.  Thus it remained until 1871, when in the aftermath of the Franco-Prussian War, the King of Prussia Wilhelm I was crowned German Emperor.  From that point forward, though the Kingdom of Prussia retained its status as a constituent state of the German Empire, all remaining Kings of Prussia also served as German Emperor, and that title took precedence.

Partitions of the Duchy of Pomerania: The Duchy of Pomerania was partitioned several times to satisfy the claims of the male members of the ruling House of Pomerania dynasty.  The partitions were named after the ducal residences: Pomerania-Barth, -Demmin, -Rügenwalde, -Stettin, -Stolp, and -Wolgast.  None of the partitions had a hereditary character, the members of the House of Pomerania inherited the duchy in common.  The duchy thus continued to exist as a whole despite its division.  The only exception was made during a war with the Margraviate of Brandenburg, when in 1338 Barnim III of Pomerania-Stettin was granted his partition as a fief directly from the Holy Roman Emperor, while Pomerania-Wolgast remained under formal Brandenburgian overlordship.  However, already in 1348, German king and later emperor Charles IV again granted the Duchy of Pomerania as a whole and the Principality of Rügen as a fief to the dukes of both Pomerania-Stettin and Pomerania-Wolgast, nullifying Brandenburg's claims by granting Imperial immediacy.

War of the Succession of Stettin: The War of the Succession of Stettin was a conflict between the Dukes of Pomerania and the Elector of Brandenburg.  It started in 1464, after the death of Duke Otto III, the last Duke of Pomerania-Stettin.  The Dukes of Pomerania-Wolgast, Eric II and Wartislaw X, held that they were Otto's rightful heirs.  Elector Frederick II of Brandenburg held that Pomerania-Stettin was a fief of Brandenburg.  Since its duke, Otto III, had died without a male heir, it should fall back to Brandenburg.

Supporting paragraphs: Partitions of the Duchy of Pomerania Charles IV, Holy Roman Emperor
Answer: House of Pomerania dynasty