Edge deletion to tree-like graph classes
Ivo Koch, Nina Pardal, Vinicius Fernandes dos Santos

TL;DR
This paper investigates the computational complexity of deleting edges to transform graphs into specific classes similar to trees, identifying cases where the problem is NP-hard and where it is tractable.
Contribution
It proves NP-hardness for deletion to cacti in bipartite graphs, and tractability for chordal graphs and quasi-threshold graphs, expanding understanding of edge deletion problems.
Findings
Deletion to cacti is NP-hard in bipartite graphs.
Deletion to chordal graphs is polynomial-time solvable.
Special algorithms are provided for quasi-threshold graphs.
Abstract
For a fixed property (graph class) , given a graph G and an integer k, the -deletion problem consists in deciding if we can turn into a graph with the property by deleting at most edges. The -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…
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Taxonomy
TopicsAdvanced Graph Theory Research · Complexity and Algorithms in Graphs · Interconnection Networks and Systems
