2003-02.fb-etal
#
Counterexamples in Chemical Ring Perception

### Abstract

Ring information is a large part of the structural topology used to
identify and characterize molecular structures. It is hence of crucial
importance to obtain this information for a variety of tasks in
computational chemistry. Many different approaches for "ring perception",
i.e., the extraction of cycles from a molecular graph, have been described.
The chemistry literature on this topic, however, reports a surprisingly
large number of incorrect statements about the properties of chemically
relevant ring sets and, in particular, about the mutual relationships of
different sets of cycles in a graph. In part these problems seem to have
arisen from a sometimes rather idiosyncratic terminology for notions that
are fairly standard in graph theory. In this contribution we translate the
definitions of concepts such as the Smallest Set of Smallest Rings,
Essential Set of Essential Rings, Extended Set of Smallest Rings, Set of
Smallest Cycles at Edges, Set of Elementary Rings, K-rings, and
beta-rings into a more widely-used mathematical language. We then
outline the basic properties of different cycle sets and provide numerous
counterexamples to incorrect claims in the published literature. These
counterexamples may have a serious practical impact because at least some
of them are molecular graphs of well-known molecules. As a consequence, we
propose a catalogue of desirable properties for chemically useful sets of
rings.

**Mathematics Subject Classification:**
05C38 (paths and cycles), 05C85 (graph algorithms)

**Keywords:**
chemical ring perception, counterexample, graph theory, cycle space, minimum length basis

Preprint available by email.

Josef.Leydold@statistik.wu-wien.ac.at