22 research outputs found
Base Size Sets and Determining Sets
Bridging the work of Cameron, Harary, and others, we examine the base size
set B(G) and determining set D(G) of several families of groups. The base size
set is the set of base sizes of all faithful actions of the group G on finite
sets. The determining set is the subset of B(G) obtained by restricting the
actions of G to automorphism groups of finite graphs. We show that for finite
abelian groups, B(G)=D(G)={1,2,...,k} where k is the number of elementary
divisors of G. We then characterize B(G) and D(G) for dihedral groups of the
form D_{p^k} and D_{2p^k}. Finally, we prove B(G) is not equal to D(G) for
dihedral groups of the form D_{pq} where p and q are distinct odd primes.Comment: 10 pages, 1 figur
Symmetry breaking in tournaments
We provide upper bounds for the determining number and the metric dimension of tournaments. A set of vertices S in 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 for an order n tournament (its determining number) is bounded by n/3, while the minimum size of a resolving set for an order n strong tournament (its metric dimension) is bounded by n/2. Both bounds are optimal.Peer ReviewedPostprint (published version
Fixing numbers for matroids
Motivated by work in graph theory, we define the fixing number for a matroid.
We give upper and lower bounds for fixing numbers for a general matroid in
terms of the size and maximum orbit size (under the action of the matroid
automorphism group). We prove the fixing numbers for the cycle matroid and
bicircular matroid associated with 3-connected graphs are identical. Many of
these results have interpretations through permutation groups, and we make this
connection explicit.Comment: This is a major revision of a previous versio
Fixing number of co-noraml product of graphs
An automorphism of a graph is a bijective mapping from the vertex set of
to itself which preserves the adjacency and the non-adjacency relations of
the vertices of . A fixing set of a graph is a set of those vertices
of which when assigned distinct labels removes all the automorphisms of
, except the trivial one. The fixing number of a graph , denoted by
, is the smallest cardinality of a fixing set of . The co-normal
product of two graphs and , is a graph having the
vertex set and two distinct vertices are adjacent if is adjacent to
in or is adjacent to in . We define a general
co-normal product of graphs which is a natural generalization of the
co-normal product of two graphs. In this paper, we discuss automorphisms of the
co-normal product of graphs using the automorphisms of its factors and prove
results on the cardinality of the automorphism group of the co-normal product
of graphs. We prove that , for
any two graphs and . We also compute the fixing number of the
co-normal product of some families of graphs.Comment: 13 page