DOI: 10.1007/s00453-004-1084-3 Algorithmica (2004) 39: 287–298 Algorithmica © 2004 Springer-Verlag NewYork, LLC Maximum Cardinality Search for Computing Minimal Triangulations of Graphs 1 Anne Berry, 2 Jean R. S. Blair, 3 Pinar Heggernes, 4 and Barry W. Peyton 5 Abstract. We present a new algorithm, called MCS-M, for computing minimal triangulations of graphs. Lex- BFS, a seminal algorithm for recognizing chordal graphs, was the genesis for two other classical algorithms: LEX M and MCS. LEX M extends the fundamental concept used in Lex-BFS, resulting in an algorithm that not only recognizes chordality, but also computes a minimal triangulation of an arbitrary graph. MCS simplifies the fundamental concept used in Lex-BFS, resulting in a simpler algorithm for recognizing chordal graphs. The new algorithm MCS-M combines the extension of LEX M with the simplification of MCS, achieving all the results of LEX M in the same time complexity. Key Words. Chordal graphs, Minimal triangulations, Minimal elimination ordering, Minimal fill. 1. Introduction. An important and widely studied problem in graph theory with ap- plications in sparse matrix computations [10], [14], [16]–[18], database management [1], [19], knowledge-based systems [11], and computer vision [7] is that of adding as few edges as possible to a given graph so that the resulting filled graph is chordal. Such a filled graph is called a minimum triangulation of the input graph. Computing a minimum triangulation is NP-hard [20]. In this paper we study the polynomially computable alter- native of finding a minimal triangulation. A minimal triangulation H of a given graph G is a triangulation such that no proper subgraph of H is a triangulation of G. Several practical algorithms exist for finding minimal triangulations [2], [5], [8], [12], [15], [18]. One such classical algorithm, called LEX M [18], is derived from the Lex-BFS (lexicographic breadth-first search) algorithm [18] for recognizing chordal graphs. 6 Both Lex-BFS and LEX M use lexicographic labels of the unprocessed vertices. As processing continues, the remaining labels grow, each potentially reaching a length proportional to the number of vertices in the graph. Lex-BFS adds to the labels of the neighbors of the vertex being processed, while LEX M adds to the labels of both neighbors and other vertices that can be reached along special kinds of paths from the 1 A preliminary version of this work appeared as [3]. This collaboration was initiated while the first two authors were visiting the University of Bergen. 2 LIMOS, UMR CNRS 6158, Universit´ e Clermont-Ferrand II, F-63177 Aubiere, France. berry@isima.fr. 3 Department of Electrical Engineering and Computer Science, United States Military Academy, West Point, NY 10996, USA. Jean.Blair@usma.edu. 4 Department of Informatics, University of Bergen, N-5020 Bergen, Norway. Pinar.Heggernes@ii.uib.no. 5 Division of Natural Sciences and Mathematics, Dalton State College, Dalton, GA 30720, USA. bpeyton@em.daltonstate.edu. 6 Lex-BFS was originally called LEX P by its authors in [18] and then later was called RTL in [19]. Received March 17, 2003; revised October 29, 2003. Communicated by H. N. Gabow. Online publication February 16, 2004.