Isolation of the diamond graph

TL;DR

This paper proves that in any large connected graph, a small vertex subset can be removed to eliminate all diamond subgraphs, with the bound on the subset size being optimal.

Contribution

It establishes a sharp bound on the size of a vertex subset needed to make a connected graph diamond-free after removal.

Findings

Existence of a small vertex subset with size at most n/5

Removal of the neighborhood of this subset yields a diamond-free graph

The bound on subset size is proven to be sharp

Abstract

A graph is $H$-free if it does not contain $H$ as a subgraph. The diamond graph is the graph obtained from $K_4$ by deleting one edge. We prove that if $G$ is a connected graph with order $n\geq 10$, then there exists a subset $S\subseteq V(G)$ with $|S|\leq n/5$ such that the graph induced by $V(G)\setminus N[S]$ is diamond-free, where $N[S]$ is the closed neighborhood of $S$. Furthermore, the bound is sharp.