Complexity of arbitrarily precolored short spiders

Determine whether Feasible Matching-Match on spiders whose legs have length at most two is polynomial-time solvable or NP-complete under arbitrary precoloring when the number of colors is unrestricted.

Background

The paper proves polynomial-time solvability for two specific precoloring regimes on spiders with legs of length at most two: precolored internal vertices adjacent to the body on length-two legs, and precolored leaves on spiders whose legs all have length exactly two. These results leave open the general case in which arbitrary vertices of such short spiders may be precolored and the number of colors is part of the input rather than fixed. The unresolved issue is whether this broader class admits a polynomial-time algorithm or is NP-complete.