Algorithmica

, Volume 75, Issue 1, pp 118–137

# Chordal Editing is Fixed-Parameter Tractable

Article

## Abstract

Graph modification problems typically ask for a small set of operations that transforms a given graph to have a certain property. The most commonly considered operations include vertex deletion, edge deletion, and edge addition; for the same property, one can define significantly different versions by allowing different operations. We study a very general graph modification problem that allows all three types of operations: given a graph Open image in new window and integers Open image in new window, and Open image in new window, the chordal editing problem asks whether Open image in new window can be transformed into a chordal graph by at most Open image in new window vertex deletions, Open image in new window edge deletions, and Open image in new window edge additions. Clearly, this problem generalizes both chordal deletion and chordal completion (also known as minimum fill-in). Our main result is an algorithm for chordal editing in time Open image in new window, where Open image in new window and Open image in new window is the number of vertices of Open image in new window. Therefore, the problem is fixed-parameter tractable parameterized by the total number of allowed operations. Our algorithm is both more efficient and conceptually simpler than the previously known algorithm for the special case chordal deletion.

### Keywords

Chordal graph Parameterized computation Graph modification problems Chordal deletion Chordal completion  Clique tree decomposition Holes Simplicial vertex sets

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