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Journal Articles Discrete Mathematics and Theoretical Computer Science Year : 2019

Computing metric hulls in graphs


We prove that, given a closure function the smallest preimage of a closed set can be calculated in polynomial time in the number of closed sets. This implies that there is a polynomial time algorithm to compute the convex hull number of a graph, when all its convex subgraphs are given as input. We then show that deciding if the smallest preimage of a closed set is logarithmic in the size of the ground set is LOGSNP-hard if only the ground set is given. A special instance of this problem is to compute the dimension of a poset given its linear extension graph, that is conjectured to be in P. The intent to show that the latter problem is LOGSNP-complete leads to several interesting questions and to the definition of the isometric hull, i.e., a smallest isometric subgraph containing a given set of vertices $S$. While for $|S|=2$ an isometric hull is just a shortest path, we show that computing the isometric hull of a set of vertices is NP-complete even if $|S|=3$. Finally, we consider the problem of computing the isometric hull number of a graph and show that computing it is $\Sigma^P_2$ complete.
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Dates and versions

hal-01612515 , version 1 (06-10-2017)
hal-01612515 , version 2 (30-07-2018)
hal-01612515 , version 3 (11-04-2019)
hal-01612515 , version 4 (23-05-2019)



Kolja Knauer, Nicolas Nisse. Computing metric hulls in graphs. Discrete Mathematics and Theoretical Computer Science, 2019, vol. 21 no. 1, ICGT 2018, ⟨10.23638/DMTCS-21-1-11⟩. ⟨hal-01612515v4⟩
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