The novel begins with the Rigonda family and two men aboard a castaway ship somewhere in the south Pacific. The family finds a hidden cache of food and they land safely on a nearby island.
After finding shelter they set out to explore the island. They soon find that everything they need seems to grow on trees. They explore, hunt wild boar, and construct a signal flag to attract passing ships.
During a tremendous storm, an emigrant ship is wrecked on the coral reef just off the island. Otto and Dominick Rigonda run to aid the passengers.
The two parties become acquainted and work together to build suitable housing. The men begin offloading supplies and constructing makeshift shelters. The women are charged with caring for the children and establishing a domicile.
Dominick and Malines come to blows, which results in Mother Lynch nominating Pauline to be queen. Pauline surprises everyone by naming Joe Binney her prime minister. A few days later, Pauline and Otto are kidnapped by natives and brought aboard their canoe. The colonists give chase and fire a volley at the natives, who allow the children to jump overboard.
A few nights later Malines and his co-conspirators are caught preparing to leave the island without the emigrants. The conspirators are marched back to the camp and put in makeshift jails. Dr. Marsh is appointed as judge over the kangaroo court.
Queen Pauline surprises the community by pardoning the criminals and offers her hand for them to kiss allegiance. For a time thereafter, there is peace, except for a group of natives hiding in the foliage who are dispatched by the gun-toting colonists.
An earthquake strikes suddenly and begins to destroy the colony, and everyone flees in the only intact boat just before the entire island sinks into the sea.
The refugees set sail for their original port, but a storm renders the ship unseaworthy once again and they must pull in to a different port. A ship bound for England is docked there and the Rigondas soon arrange for their passage home to England. The novel ends as the children are finally reunited with their parents, who are none too eager to hear their tales of adventure.

Question: What is the name of the family in the story?
Answer: The Rigonda family.
Alternative answer: Rigonda