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We consider a graph-theoretic elimination process which is related to performing Gaussian elimination on sparse symmetric positive definite systems of linear equations. We give a new linear-time algorithm to cal culate the fill-in produced by any elimination ordering, and we give two new related algorithms for finding orderings with small fill-in. One algorithm, based on breadth-first search, finds a perfect elimination ordering, if any exists, in 0(n+e) time, if the problem graph has n vertices and e edges. An extension of this algorithm finds a minimal (but not necessarily minimum) ordering in 0(ne) time. We conjecture that the problem of finding a minimum ordering is NP-complete.

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A graph is an interval graph tf and only if each of Its verttces can be associated with an interval on the real hne m such a way that two vertices are adjacent m the graph exactly when the corresponding mtervals have a nonempty mtersectmn An effictent algonthrn for testing tsomorpinsm of interval graphs ts unplemented using a data structure called a PQ-tree. The algorithm runs m O(n + e) steps for graphs having n vemces and e edges It is shown that for a somewhat larger class of graphs, namely the chordal graphs, lsomorpinsm is as hard as for general graphs

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