Text this: Existence of independent [1, 2]-sets in caterpillars

  ______    ______    _____    __   __    _  _   
 /_   _//  /_   _//  /  ___||  \ \\/ //  | \| || 
 `-| |,-    -| ||-  | // __     \ ` //   |  ' || 
   | ||     _| ||_  | \\_\ ||    | ||    | .  || 
   |_||    /_____//  \____//     |_||    |_|\_|| 
   `-`'    `-----`    `---`      `-`'    `-` -`