Abstract
The reduction number r(G) of a graph G is the maximum integer m≤|E(G)| such that the graphs G−E, E⊆E(G),|E|≤m, are mutually non-isomorphic, i.e., each graph is unique as a subgraph of G. We prove that 
and show by probabilistic methods that r(G) can come close to this bound for large orders. By direct construction, we exhibit graphs with large reduction number, although somewhat smaller than the upper bound. We also discuss similarities to a parameter introduced by Erdős and Rényi capturing the degree of asymmetry of a graph, and we consider graphs with few circuits in some detail.
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Supported by a grant from the Danish Natural Science Research Council.
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Andersen, L., Vestergaard, P. & Tuza, Z. Largest Non-Unique Subgraphs. Graphs and Combinatorics 22, 453–470 (2006). https://doi.org/10.1007/s00373-006-0676-x
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DOI: https://doi.org/10.1007/s00373-006-0676-x
Keywords
- Asymmetry
- Edge deletion
- Isomorphism
- Random graph
- Reduction number
- Spanning subgraph
- Symmetry
- Tree
- Unicyclic graph
- Unique subgraph