Medial Symmetry Type Graphs

Medial Symmetry Type Graphs

A k-orbit map is a map with its automorphism group partitioning the set of flags into k orbits. Recently k-orbit maps were studied by Orbanić, Pellicer and Weiss, for k 6 4. In this paper we use symmetry type graphs to extend such study and classify all the types of 5-orbit maps, as well as all self-dual, properly and improperly, symmetry type of k-orbit maps with k 6 7. Moreover, we determine,...

Symmetry Type Graphs of Polytopes and Maniplexes

A k-orbit maniplex is one that has k orbits of flags under the action of its automorphism group. In this paper we extend the notion of symmetry type graphs of maps to that of maniplexes and polytopes and make use of them to study k-orbit maniplexes, as well as fully-transitive 3-maniplexes. In particular, we show that there are no fully-transtive korbit 3-mainplexes with k > 1 an odd number, we...

Symmetry Breaking in Graphs

A labeling of the vertices of a graph G, φ : V (G) → {1, . . . , r}, is said to be r-distinguishing provided no automorphism of the graph preserves all of the vertex labels. The distinguishing number of a graph G, denoted by D(G), is the minimum r such that G has an r-distinguishing labeling. The distinguishing number of the complete graph on t vertices is t. In contrast, we prove (i) given any...

Generalized symmetry of graphs

Symmetry of graphs has been extensively studied over the past 50 years by using automorphisms of graphs and group theory which have played an important role in graph, and promising and interesting results have been obtained[2,20,21]. We introduced generalized symmetry of graphs and investigated it by using endomorphisms of graphs and semigroup theory. In this paper, I will survey some results w...

Visualisation of Symmetry in Graphs