Nonuniform symmetry breaking in noncommutativeλΦ4theory

Symmetry Breaking in Tournaments

We provide upper bounds for the determining number and the metric dimension of tournaments. A set of vertices S ⊆ V (T ) is a determining set for a tournament T if every nontrivial automorphism of T moves at least one vertex of S, while S is a resolving set for T if every two distinct vertices in T have different distances to some vertex in S. We show that the minimum size of a determining set ...

Symmetry Breaking

Symmetry occurs in many problems in aritifical intelligence. For example, in the n-queens problem, the chessboard can be rotated 90◦. As a second example, several machines in a factory might have the same capacity. In a production schedule, we might therefore be able to swap the jobs on machines with the same capacity. As a third example, two people within a company might have the same skills. ...

Symmetry Breaking

Breaking symmetry in myxobacteria

Multicellular animals and plants develop from genetically equivalent cells, yet neighboring cells often adopt different fates. Some well-studied examples are the formation of sensory bristles from a sheet of uniform neuroectodermal cells in Drosophila, and the formation of hair or feathers in the uniform skin ectoderm of vertebrates. This breaking of symmetry within groups of equivalent cells d...

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...