In this paper two methods for automatic generation of connected chordal graphs are proposed: the first one is based on new results concerning the dynamic maintenance of chordality under edge insertions; the second is based on expansion/merging of maximal cliques. Theoretical and experimental results are presented. In both methods, chordality is preserved along the whole generation process.In the solution of algorithmic problems, graphs can play different roles, being the very input for an algorithm or simply an auxiliary data structure handled by it. In the first case, the generation of suitable instances (i.e. input graphs satisfying given constraints) can be so complex that it constitutes a further problem, sometimes as hard to solve as the original one.Chordal graphs are a broadly studied class, as their peculiar clique-based structure allows the solution of many algorithmic problems. See, for instance, Chandran et al. (2003), Gavril (1972, and Kumar and Madhavan (2002). Since the generation of instances for testing these algorithms is often required, our goal in this paper is to develop procedures for automatically constructing large connected chordal graphs, with more than 10,000 vertices.Rather than generating connected graphs at random and testing whether they are chordal or not, we focus here on generation procedures in which graphs are constructed while chordality is maintained during the whole generation process. Two methods are presented.The incremental method is based on new results about the dynamic maintenance of chordality under edge insertions developed by Araujo (2004). It allows the generation of L. Markenzon's research is partially supported by grant 301068/2003-8, CNPq, Brazil. L. Markenzon ( ) · O. Vernet

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The set of minimal vertex separators of chordal graphs is usually obtained by two-phase algorithms. Based on properties of the lexicographic breadth-first search, we propose a new one-phase algorithm. We present also a characterization for planar chordal graphs; using our proposed algorithm as an initial step, the implementation of the recognition algorithm becomes trivial.

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