Abstract
Determining vertex subsets are known tools to provide information about automorphism groups of graphs and, consequently about symmetries of graphs. In this paper, we provide both lower and upper bounds of the minimum size of such vertex subsets, called the determining number of the graph. These bounds, which are performed for arbitrary graphs, allow us to compute the determining number in two different graph families such are cographs and unit interval graphs.
Highlights
Introduction and PreliminariesThe graph isomorphism problem is not known to be solvable in polynomial time nor to be NP-complete and it is well known that constructing the automorphism group is at least as difficult as solving the graph isomorphism problem
We present the announced lower bound of the determining number of a graph, in terms of the corresponding parameter of its twin graph
The main tool that we used is the twin graph, defined in [22] to study the metric dimension of graphs, and which has proven to be useful for obtaining determining sets and for computing the determining number
Summary
The graph isomorphism problem is not known to be solvable in polynomial time nor to be NP-complete (see [1]) and it is well known that constructing the automorphism group is at least as difficult (in terms of computational complexity) as solving the graph isomorphism problem (see [2]). We present the general behaviour of the stabilizer of a vertex subset under the mapping Te and the special situation of plenty twin sets. Assume that S is a plenty twin set and let ψ ∈ StabG1 (S1 ), and let us construct the mapping φ : V ( G ) −→ V ( G ) in the following way. This means that StabG (S ∪ Ω) = StabG (S) ∩ StabG (Ω) = {idG } and so S ∪ Ω is a determining set of G This gives the desired bound, since |S| = Det( Gr ) and |Ω| = n − n(1). It is proved in [14] that a twin-free graph has determining number at most the half of its order
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