Abstract
Chordal graphs, also called triangulated graphs, are important in algorithmic graph theory. In this paper we generalise the definition of chordal graphs to the class of directed graphs. Several structural properties of chordal graphs that are crucial for algorithmic applications carry over to the directed setting, including notions like simplicial vertices, perfect elimination orderings, and characterisation by forbidden subgraphs resembling chordless cycles. Moreover, just as chordal graphs are related to treewidth, the chordal digraphs will be related to Kelly-width. Chordal digraphs coincide with the perfect elimination digraphs arising in the study of Gaussian elimination on sparse linear systems [Haskins and Rose, 1973].
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