1-Extendability of Independent Sets

In the 70 s, 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 .