On The Fixatic Number of Graphs

Abstract

The fixing number of a graph G is the smallest cardinality of a set of vertices F ⊆ V (G) such that only the trivial automorphism of G fixes every vertex inThe cardinality of a largest fixatic partition is called the fixatic number of G. In this paper, we study the fixatic numbers of graphs. Sharp bounds for the fixatic number of graphs in general and exact values with specified conditions are given. Some realizable results are also given in this paper. N(u) − {v} = N(v) − {u}, then u and v are called twin vertices (or simply twins) in G. If for a vertex u of G, there exists a vertex v = u in G such that u, v are twins in G, then u is said to be a twin in G. A set T ⊆ V (G) is said to be a twin-set in G if any two of its elements are twins.An automorphism α of G, α : V (G) → V (G), is a bijective mapping such that α(u)α(v) ∈ E(G) if and only if uv ∈ E(G). Thus, each automorphism α of G is a permutation of the vertex set V (G) which preserves adjacencies and non-adjacencies. The automorphism group of a graph G, denoted by Γ(G), is the set of all automorphisms of a graphFor a vertex v of a graph G, the orbit of v, denoted O(v), is the set of all vertices {u ∈ V (G) : α(v) = u for some α ∈ Γ(G)}. Two vertices u and v are similar if they belong to the same orbit.A vertex v is fixed by a group element gis trivial. The fixing number of a graph G, denoted by f ix(G), is the smallest cardinality of a fixing set of G [8]. The graphs