Edge deletion to tree-like graph classes

Ivo Koch, Nina Pardal, Vinicius Fernandes Dos santos

Research output: Contribution to journalArticlepeer-review

Abstract

For a fixed property (graph class) II, given a graph G and an integer k, the II-deletion problem consists in deciding if we can turn G into a graph with the property II by deleting at most k edges. The II-deletion problem is known to be NP-hard for most of the well-studied graph classes, such as chordal, interval, bipartite, planar, comparability and permutation graphs, among others; even deletion to cacti is known to be NP-hard for general graphs. However, there is a notable exception: the deletion problem to trees is polynomial. Motivated by this fact, we study the deletion problem for some classes similar to trees, addressing in this way a knowledge gap in the literature. We prove that deletion to cacti is hard even when the input is a bipartite graph. On the positive side, we show that the problem becomes tractable when the input is chordal, and for the special case of quasi-threshold graphs we give a simpler and faster algorithm. In addition, we present sufficient structural conditions on the graph class II that imply the NP-hardness of the II-deletion problem, and show that deletion from general graphs to some well-known subclasses of forests is NP-hard.
Original languageEnglish
Pages (from-to)122-131
Number of pages10
JournalDiscrete Applied Mathematics
Volume348
Early online date2 Feb 2024
DOIs
Publication statusPublished - 15 May 2024
Externally publishedYes

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