Polynomial enumeration of chordless cycles on cyclically orientable graphs

TL;DR

This paper introduces an efficient algorithm for enumerating all chordless cycles in cyclically orientable graphs, with each cycle found exactly once in quadratic time relative to the number of vertices.

Contribution

The paper presents a novel algorithm that efficiently enumerates chordless cycles in cyclically orientable graphs, improving over existing methods by ensuring unique cycle detection and quadratic time complexity.

Findings

Algorithm finds each chordless cycle only once

Time complexity is $ ext{O}(n^2)$ where n is vertices

Applicable to cyclically orientable graphs

Abstract

In a finite undirected simple graph, a chordless cycle is an induced subgraph which is a cycle. A graph is called cyclically orientable if it admits an orientation in which every chordless cycle is cyclically oriented. We propose an algorithm to enumerate all chordless cycles of such a graph. Compared to other similar algorithms, the proposed algorithm have the advantage of finding each chordless cycle only once in time complexity $\mathcal{O}(n^2)$ in the input size, where $n$ is the number of vertices.