Distinguishing graphs via cycles

Abstract

In this paper, we employ the cycle regularity parameter to devise efficient recognition algorithms for three highly symmetric graph families: folded cubes, $I$-graphs, and double generalized Petersen graphs.

For integers $\ell ,\lambda ,m$ a simple graph is $[\ell ,\lambda ,m]$-cycle regular if every path of length $\ell$ belongs to exactly $\lambda$ different cycles of length $m$. We identify all $[1,\lambda ,8]$-cycle regular $I$-graphs and all $[1,\lambda ,8]$-cycle regular double generalized Petersen graphs. For $n\ge 7$ we show that a folded cube $F{Q}_n$ is $[1,n-1,4]$, $[1,4n^2-12n+8,6]$ and $[2,4n-8,6]$-cycle regular, and identify the corresponding exceptional values of cycle regularity for $n<7$. As a consequence, we describe a linear recognition algorithm for double generalized Petersen graphs, an $O(\left|E\right|log\left|V\right|)$ recognition algorithm for the family of folded cubes, and an $O\left({\left|V\right|}^2\right)$ recognition algorithm for $I$-graphs.

We believe the structural observations and methods used in the paper are of independent interest and could be used to solve other algorithmic problems. The results of this paper have been presented at COCOON 2021.