ATOMS OF CYCLIC EDGE CONNECTIVITY IN TETRAVALENT GRAPHS

A cyclic edge-cut of a connected graph G is an edge set, the removal of which separate two cycles. If G has a cyclic edge-cut, then it is called cyclically separable. An induced subgraph P of G is called a cyclic part of G if there exists a cyclic edge-cut S such that P is a component of G — S. A cyclic part minimal under inclusion is called an atom. Also a cyclic part with minimum cardinality is called proper atom. For a cyclically separable graph G, the cyclic edge connectivity of a graph G, denoted by Ac(G), is the minimal cardinality over all cyclic edge cuts. In this article, we establish some properties of atoms and cyclic parts in tetravalent graphs.