On minimal N+– rank graphs

Lipták and Tunçel in [Lipták L. and Tunçel L. The stable set problem and the lift-and-project ranks of graphs, Mathematical Programming B 98 (2003) 319–353] study the minimum number of nodes needed in a graph to have N + – rank k, proving that it is at least 3k. They conjecture that, for any k, there is a k–minimal graph, i.e. a graph with 3k nodes and N + – rank k. They find one 2–minimal subdivision of K4 and conjecture that there is always a k–minimal subdivision of K k + 2 . In this paper, we present some necessary conditions for a graph to be k–minimal. Using these conditions we prove that Lipták and Tunçelʹs first conjecture holds for k = 3 showing a 3–minimal graph which is a subdivision of K5. However, from the complete characterization of 2−minimal graphs and the necessary conditions, we prove that the second conjecture is not correct showing a stronger result: for k > 4 there is not a k–minimal subdivision of any complete graph. er to analyze the existence of k–minimal graphs for k ⩾ 4 , results on the behavior of ranks under subdivisions in a graph become useful. In this way, in connection with the graph classes B 0 and B presented by Lipták and Tunçel, we define a new class B + that, in particular, contains the known k–minimal graphs. We obtain a characterization of these three classes in terms of the disjunctive rank, allowing us to give a simpler proof of the rank invariance under graph odd subdivisions in these classes.