Symmetry properties of chordal rings of degree 3

Abstract

This paper presents the main properties of chordal rings of degree 3. This family of graphs is strongly related to circulant graphs, which are actually often called chordal rings too. The use of triangles in the plane to represent the vertices allows one to associate a plane tessellation to every chordal ring. By using this geometrical approach, we study the recognition and the isomorphism problems for this class of graphs. A polynomial-time algorithm to recognize chordal rings and a polynomial-time algorithm to decide isomorphism between two chordal rings, given by its adjacency list, are presented. Both algorithms are based on the study of the 4- and 6-cycles of the graph. This approach is also applied to the characterization of the automorphisms-group of chordal rings. We believe that these results produce useful tools for further works as, in particular, the study of compact routing schemes, and the study of optical routing protocols in edge-transitive chordal rings.