1-Extendability of independent sets

Pierre Bergé; Anthony Busson; Carl Feghali; Rémi Watrigant

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arXiv · 2204.05809

1-Extendability of independent sets

Pierre Bergé·Anthony Busson·Carl Feghali·Rémi Watrigant

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Abstract

In the 70s, Berge introduced 1-extendable graphs (also called B-graphs), which are graphs where every vertex belongs to a maximum independent set. Motivated by an application in the design of wireless networks, we study the computational complexity of 1-extendability, the problem of deciding whether a graph is 1-extendable. We show that, in general, 1-extendability cannot be solved in $2^{o(n)}$ time assuming the Exponential Time Hypothesis, where $n$ is the number of vertices of the input graph, and that it remains NP-hard in subcubic planar graphs and in unit disk graphs (which is a natural model for wireless networks). Although 1-extendability seems to be very close to the problem of finding an independent set of maximum size (a.k.a. Maximum Independent Set), we show that, interestingly, there exist 1-extendable graphs for which Maximum Independent Set is NP-hard. Finally, we investigate a parameterized version of 1-extendability.

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Pierre Bergé, Anthony Busson, Carl Feghali, Rémi Watrigant. 2022-04-12. 1-Extendability of independent sets. https://arxiv.org/abs/2204.05809

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