Extending partial representations of subclasses of chordal graphs

Chordal graphs are intersection graphs of subtrees of a tree T. We investigate the complexity of the partial representation extension problem for chordal graphs. A partial representation specifies a tree T′ and some pre-drawn subtrees of T′. It asks whether it is possible to construct a representation inside a modified tree T which extends the partial representation (i.e., keeps the pre-drawn subtrees unchanged). We consider four modifications of T′ leading to vastly different problems: (i) T=T′, (ii) T is a subdivision of T′, (iii) T is a supergraph of T′, and (iv) T′ is a topological minor of T. In some cases, it is interesting to consider the complexity even when 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. We further study the parametrized complexity of the problems when parametrized by the number of pre-drawn subtrees, the number of components of the input graph G and the size of the tree T′. We describe an interesting relation with integer partition problems. The problem 3-Partition is used for all NP-completeness reductions. When the space in T′ is limited, partial representation extension of proper interval graphs is “equivalent” to the BinPacking problem.