Let G=(V,E) be a simple connected graph and 08a18add02451d8e1d2a1187c1de3e" title="Click to view the MathML source">S={w1,…,wt}⊆V an ordered subset of vertices. The metric representation of a vertex u ∈ V with respect to S is the t -vector a055e38fd9ec3a33cd" title="Click to view the MathML source">r(u|S)=(dG(u,w1),…,dG(u,wt)), where dG(u, v ) represents the length of a shortest u−v path in G. We call S a k-antiresolving set if k is the largest positive integer such that for every vertex v∈V−S there exist other k−1 different vertices v1,…,vk−1∈V−S with r(v|S)=r(v1|S)=鈰?r(vk−1|S). The k-metric antidimension of G is the minimum cardinality among all the k-antiresolving sets for G.
We address the k-metric antidimension problem by proposing a true-biased algorithm with success rate above 80% when considering random graphs of size at most 100. The proposed algorithm is used to determine the privacy guarantees offered by two real-life social graphs with respect to (k, 鈩?-anonymity. We also investigate theoretical properties of the k-metric antidimension of graphs. In particular, we focus on paths, cycles, complete bipartite graphs and trees.