Deleting to Structured Trees

TL;DR

This paper studies the problem of deleting vertices or edges to form a full binary tree, proving NP-hardness and fixed-parameter tractability, highlighting differences from related polynomial-time solvable problems.

Contribution

It introduces a new variant of the Feedback Vertex Set problem focused on full binary trees, establishing NP-hardness and FPT results.

Findings

Both vertex and edge deletion problems are NP-hard.

Both problems are fixed-parameter tractable (FPT).

Contrasts with polynomial-time solvability of related problems.

Abstract

We consider a natural variant of the well-known Feedback Vertex Set problem, namely the problem of deleting a small subset of vertices or edges to a full binary tree. This version of the problem is motivated by real-world scenarios that are best modeled by full binary trees. We establish that both versions of the problem are NP-hard, which stands in contrast to the fact that deleting edges to obtain a forest or a tree is equivalent to the problem of finding a minimum cost spanning tree, which can be solved in polynomial time. We also establish that both problems are FPT by the standard parameter.