Discrete Mathematics & Theoretical Computer Science |

- 1 Laboratoire d'Informatique et Systèmes [LIS]
- 2 Laboratoire d'Informatique et des Systèmes
- 3 Laboratoire d'Informatique et des Systèmes (LIS) (Marseille, Toulon)
- 4 Combinatorics, Optimization and Algorithms for Telecommunications

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.

Source: HAL:hal-01612515v4

Volume: vol. 21 no. 1, ICGT 2018

Published on: May 23, 2019

Accepted on: May 15, 2019

Submitted on: July 30, 2018

Keywords: [INFO.INFO-CC]Computer Science [cs]/Computational Complexity [cs.CC]

Funding:

- Source : OpenAIRE Graph
*Théorie métrique des graphes*; Funder: French National Research Agency (ANR); Code: ANR-17-CE40-0015*Graphes, Algorithmes et TOpologie*; Funder: French National Research Agency (ANR); Code: ANR-16-CE40-0009

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