Removing Symmetry in Circulant Graphs and Point-Block Incidence Graphs

An automorphism of a graph is mapping the vertices onto themselves such that connections between respective edges are preserved. A vertex v in G fixed if it mapped to itself under every G. The fixing number minimum vertices, when fixed, fixes all determination numbers important as can be useful determining group automorphisms graph-a famous and difficult problem. Fixing were introduced initially studied by Gibbons Laison, Erwin Harary Boutin. In this paper, we investigate for graphs with an underlying cyclic structure, which provides inherent presence symmetry. We first determine circulant graphs, showing many cases 2. However, also show twins, pairs same neighbourhoods, have considerably higher numbers. This paper investigates point-block incidence lie at intersection theory combinatorial design theory. present surprising result-identifying infinite families any vertex, thus removing symmetries from graph.