On the Edit Distance of Powers of Cycles

TL;DR

This paper determines the edit distance function for graphs avoiding powers of cycles, providing exact formulas for large cycle lengths and partial results for smaller ones, advancing understanding of graph edit distances.

Contribution

It explicitly characterizes the edit distance function for forbidding powers of cycles, a problem previously unresolved for many parameters.

Findings

Exact edit distance functions for large cycle powers when h ≥ 2t(t+1)+1 and h not divisible by t+1.

Partial formulas for the case when h ≥ 2t(t+1)+1 and h divisible by t+1.

Results for smaller cycle lengths h, extending the known cases.

Abstract

The edit distance between two graphs on the same labeled vertex set is defined to be the size of the symmetric difference of the edge sets. The edit distance function of a hereditary property $\mathcal{H}$ is a function of $p\in [0,1]$ that measures, in the limit, the maximum normalized edit distance between a graph of density $p$ and $\mathcal{H}$. In this paper, we address the edit distance function for $\mbox{Forb}(H)$, where $H=C_h^t$, the $t^{\rm th}$ power of the cycle of length $h$. For $h\geq 2t(t+1)+1$ and $h$ not divisible by $t+1$, we determine the function for all values of $p$. For $h\geq 2t(t+1)+1$ and $h$ divisible by $t+1$, the function is obtained for all but small values of $p$. We also obtain some results for smaller values of $h$.