What is the fewest number of cardinal electors are needed to validly elect a pope? What is the fewest number of cardinal electors are needed to validly elect a pope?

Pope St. John Paul II, in his Apostolic Constitution Universi Dominici Gregis on the vacancy of the Apostolic See and the election  of the Roman Pontiff has this to say on the electors of a papal election, but does not provide us with the minimum number of cardinal electors needed to validly choose a pope:


  
  The right to elect the Roman Pontiff belongs exclusively to the Cardinals of Holy Roman Church, with the exception of those who have reached their eightieth birthday before the day of the Roman Pontiff's death or the day when the Apostolic See becomes vacant. The maximum number of Cardinal electors must not exceed one hundred and twenty. The right of active election by any other ecclesiastical dignitary or the intervention of any lay power of whatsoever grade or order is absolutely excluded.
  


What happens if the conclave is to be held during a time of war or some other event and the possible number of cardinal elector at Rome dwindles to 1, 2, 3 or 4. Is there a minimum number of cardinals needed to produce a sovereign pontiff? 

We know that conclave of 1277 had the fewest number of cardinals in a papal election:


  The papal election of 1277 (May 30 – November 25), convened in Viterbo after the death of Pope John XXI, was the smallest papal election since the expansion of suffrage to cardinal-priests and cardinal-deacons, with only seven cardinal electors (following the deaths of three popes who had not created cardinals).3 Because John XXI had revoked Ubi periculum, the papal bull of Pope Gregory X establishing the papal conclave, with his own bull Licet felicis recordationis, the cardinal electors were able to take their time. After six months of deliberation, the cardinals eventually elected their most senior member Giovanni Gaetano Orsini as Pope Nicholas III. From the end of the election until Nicholas III's first consistory on March 12, 1278, the number of living cardinals—seven—was the lowest in the history of the Roman Catholic Church. - Papal election, 1277


I have heard about the possibility that 4 cardinal electors would be allowed, but I can not find any documents to support such a number.
There is no minimum number of cardinals set in the apostolic constitution Universi Dominici Gregis, written in 1996 by Pope St. John Paul II. This document is the current norm for papal elections. It prescribes such things as the length of the dead pope's funeral, the procedure for managing Vatican City and the Diocese of Rome while the See is vacant, and the procedure and timetable for setting up the conclave to elect the next pope.
There are groups, or at least one rotating group, of cardinals, called the &quot;Particular Congregations&quot;, which oversee day-to-day operations of the Vatican:

The Particular Congregation is made up of the Cardinal Camerlengo of [the] Holy Roman Church and three Cardinals, one from each order [that is, a Cardinal Deacon, Cardinal Priest, and Cardinal Bishop]. ... The office of these Cardinals, called Assistance, ceases at the conclusion of the third full day, and their place is taken by others.
The Particular Congregations are to deal only with questions of lesser importance which arise on a daily basis or from time to time.
(Universi Dominici Gregis, sections 7–8)

Perhaps this is where your &quot;four cardinals&quot; idea comes from.
The apostolic constitution specifies that all the cardinals who are not &quot;legitimately impeded&quot; (section 7) must come to the conclave as soon as they are informed of the pontiff's death. No further mention is made specifically of any number required in order to cast a valid vote for the pope. Presumably this is to allow an election regardless of how many cardinals are able to make it to Rome.