The Proper interval colored graph problem for caterpillar trees

Abstract

This paper studies the computational complexity of the Proper Interval Colored Graph problem (PICG), when the input graph is a colored tree. We show that the problem is hard for the class of caterpillar trees. To prove our result we make use of a close relationship between intervalizing problems and graph layout problems. We define a graph layout problem the Proper Colored Layout Problem (PCLP). Although PCLP is not equivalent to PICG, by transforming the input graph we will stablish a close relationship between both problems. The main result is that the PICG is NP-complete for colored caterpillars of hair length 2 and in P for caterpillars of hair length 1 or 0.