The VC-Dimension of Graphs with Respect to k-Connected Subgraphs

TL;DR

This paper investigates the VC-dimension of graph-induced set systems based on k-connected subgraphs, providing bounds, complexity results, and efficient algorithms for special graph classes.

Contribution

It establishes tight bounds for the VC-dimension, proves NP-completeness for computing it in general and specific subclasses, and offers linear-time algorithms for graphs with bounded clique-width.

Findings

Tight upper and lower bounds for VC-dimension.

NP-completeness of computing VC-dimension in general and certain subclasses.

Linear-time decision algorithm for graphs with bounded clique-width.

Abstract

We study the VC-dimension of the set system on the vertex set of some graph which is induced by the family of its $k$-connected subgraphs. In particular, we give tight upper and lower bounds for the VC-dimension. Moreover, we show that computing the VC-dimension is $\mathsf{NP}$-complete and that it remains $\mathsf{NP}$-complete for split graphs and for some subclasses of planar bipartite graphs in the cases $k = 1$ and $k = 2$. On the positive side, we observe it can be decided in linear time for graphs of bounded clique-width.