The edit distance function of some graphs

TL;DR

This paper calculates the maximum asymptotic edit distance between certain graphs and hereditary properties, extending previous work to new classes of graphs like specific cycles, their modifications, and paths.

Contribution

It determines the edit distance functions for new classes of graphs, including modified cycles and paths, using methods from prior research.

Findings

Calculated the edit distance function for $C_8^{*}$.

Determined the edit distance function for $ ilde{C}_n$.

Established the edit distance function for $P_n$.

Abstract

The edit distance function of a hereditary property $\mathscr{H}$ is the asymptotically largest edit distance between a graph of density $p\in[0,1]$ and $\mathscr{H}$. Denote by $P_n$ and $C_n$ the path graph of order $n$ and the cycle graph of order $n$, respectively. Let $C_{2n}^*$ be the cycle graph $C_{2n}$ with a diagonal, and $\widetilde{C_n}$ be the graph with vertex set $\{v_0, v_1, \ldots, v_{n-1}\}$ and $E(\widetilde{C_n})=E(C_n)\cup \{v_0v_2\}$. Marchant and Thomason determined the edit distance function of $C_6^{*}$. Peck studied the edit distance function of $C_n$, while Berikkyzy et al. studied the edit distance of powers of cycles. In this paper, by using the methods of Peck and Martin, we determine the edit distance function of $C_8^{*}$, $\widetilde{C_n}$ and $P_n$, respectively.