We consider the following problem, called Relative Minimal Elimination Ordering. Given a graph G = (V; E) which is a subgraph of the chordal graph G 0 = (V; E 0 ), compute an inclusion minimal chordal graph G 00 = (V; E 00 ), such that E ` E 00 ` E 0 . We show that this can be done in O(nm) time. This extends the results of [2]. The algorithm is based only on well known results on chordal graphs.

We show that for graphs of bounded degree, a subset minimal elimination ordering can be determined in almost linear time.

We prove that the problem to get an inclusion minimal elimination ordering can be solved in linear time for planar graphs. The basic structure of the linear time algorithm is as follows. We select a vertex r as maximum and get a first approximation of a minimal elimination ordering considering a vertex x as smaller than y if x has a larger distance than y from r. Using planarity, one can determine the fill-in edges joining two vertices of the same distance from r almost immediately. The algorithm determines an O(n)-representation of these fill-in edges. To determine the final fill-in ordering, we use similar techniques as in the general parallel minimal elimination algorithm of [5].

In this paper we study the computational complexity (both sequential and parallel) of the maximum matching problem for chordal and strongly chordal graphs. We show that there is a linear time greedy algorithm for a maximum matching in a strongly chordal graph provided a strongly perfect elimination ordering is known. This algorithm can be also turned into a parallel algorithm. The technique used can be also extended for the multidimensional matching for chordal and strongly chordal graphs yielding the first polynomial time algorithms for these classes of graphs (the multidimensional matching is NPcomplete in general).

We show that a minimum fill-in ordering of a graph can be determined in linear time if it can be modularly decomposed into chordal graphs. This generalizes results of [2]. We show that the treewidth of these graphs can be determined in O((n +m) log n) time.

We present an alternative linear time algorithm that computes an ordering that produces a fill-in that is minimal with respect to the subset relation. It is simpler than the algorithm in [6] and is easily parallelizable. The algorithm does not rely on the computation of a breadth-first search tree.

We prove that lexicographic breadth-first search is P-complete and that a variant of the parallel perfect elimination procedure of P. Klein [24] is powerful enough to compute a semi-perfect elimination ordering in sense of [23] if certain induced subgraphs are forbidden. We present an efficient parallel breadth first search algorithm for all graphs which have no cycle of length greater four and no house as an induced subgraph. A side result is that a maximal clique can be computed in polylogarithmic time using a linear number of processors.