Degenerate Vertex Cuts in Sparse Graphs

TL;DR

This paper investigates the size and existence of $k$-degenerate vertex cuts in sparse graphs, establishing bounds and conditions under which such cuts exist or are absent.

Contribution

It provides new bounds on the size of graphs without $k$-degenerate cuts and characterizes conditions for the existence of minimum $k$-degenerate cuts in connected graphs.

Findings

Graphs without $k$-degenerate cuts have size at least  rac{1}{2}(k+\u03a9(\u221a{k}))n.

Graphs of order at least 5 without a 2-degenerate cut have size at least  rac{27n-35}{10}.

Connected graphs with certain size bounds have a minimum $k$-degenerate cut.

Abstract

For a non-negative integer $k$, a vertex cut in a graph is $k$-degenerate if it induces a $k$-degenerate subgraph. We show that a graph of order $n$ at least $2k+2$ without a $k$-degenerate cut has the size at least $\frac{1}{2}\left(k+\Omega\left(\sqrt{k}\right)\right)n$ and that a graph of order $n$ at least $5$ without a $2$-degenerate cut has the size at least $\frac{27n-35}{10}$. For $k\geq 2$, we show that a connected graph $G$ of order $n$ at least $k+6$ and size $m$ at most $\frac{k+3}{2}n+\frac{k-1}{2}$ has a minimum $k$-degenerate cut.