A Complexity Analysis of the c-Closed Vertex Deletion Problem

TL;DR

This paper analyzes the computational complexity of the c-Closed Vertex Deletion problem, exploring its NP-hardness, kernelization bounds, and fixed-parameter tractability on specific graph classes.

Contribution

It provides new complexity results, kernel size bounds, and algorithms for c-Closed Vertex Deletion, including NP-hardness proofs and fixed-parameter tractability on certain graph classes.

Findings

NP-hardness for bipartite graphs with bounded degree

Kernel size bounds based on bad pairs parameter

Polynomial-time solution on certain interval graphs

Abstract

A graph is $c$-closed when every pair of nonadjacent vertices has at most $c-1$ common neighbors. In $c$-Closed Vertex Deletion, the input is a graph $G$ and an integer $k$ and we ask whether $G$ can be transformed into a $c$-closed graph by deleting at most $k$ vertices. We study the classic and parameterized complexity of $c$-Closed Vertex Deletion. We obtain, for example, NP-hardness for the case that $G$ is bipartite with bounded maximum degree. We also show upper and lower bounds on the size of problem kernels for the parameter $k$ and introduce a new parameter, the number $x$ of vertices in bad pairs, for which we show a problem kernel of size $\mathcal{O}(x^3 + x^2\cdot c))$. Here, a pair of nonadjacent vertices is bad if they have at least $c$ common neighbors. Finally, we show that $c$-Closed Vertex Deletion can be solved in polynomial time on unit interval graphs with depth at most $c+1$ and that it is fixed-parameter tractable with respect to the neighborhood diversity of $G$.