Extending partial representations of subclasses of chordal graphs

Abstract

Chordal graphs are intersection graphs of subtrees in a tree. We investigate complexity of the partial representation extension problem for chordal graphs. A partial representation specifies a tree T′ and some pre-drawn subtrees. It asks whether it is possible to construct a representation inside a modified tree T which extends the partial representation (keeps the pre-drawn subtrees unchanged). We consider four modifications of T′ and get vastly different problems. In some cases, the problem is interesting even if just T′ is given and no subtree is pre-drawn. Also, we consider three well-known subclasses of chordal graphs: Proper interval graphs, interval graphs and path graphs. We give an almost complete complexity characterization. In addition, we study parametrized complexity by the number of predrawn subtrees, the number of components and the size of the tree T′. We describe an interesting relation with integer partition problems. The problem3-PARTITION is used in theNP-completeness reductions. The BINPACKING problem is closely related to the extension of interval graphswhen space in T′ is limited, and we obtain "equivalency" with BINPACKING. © Springer-Verlag 2012.