The 0–1 inverse maximum stable set problem

Abstract

In this paper we study the 0–1 inverse maximum stable set problem, denoted by I S { 0 , 1 } . Given a graph and a fixed stable set, it is to delete the minimum number of vertices to make this stable set maximum in the new graph. We also consider I S { 0 , 1 } against a specific algorithm such as G r e e d y and 2 o p t , aiming to delete the minimum number of vertices so that the algorithm selects the given stable set in the new graph; we denote them by I S { 0 , 1 } , g r e e d y and I S { 0 , 1 } , 2 o p t , respectively. Firstly, we show that they are NP-hard, even if the fixed stable set contains only one vertex. Secondly, we achieve an approximation ratio of 2 − Θ ( 1 l o g Δ ) for I S { 0 , 1 } , 2 o p t . Thirdly, we study the tractability of I S { 0 , 1 } for some classes of perfect graphs such as comparability, co-comparability and chordal graphs. Finally, we compare the hardness of I S { 0 , 1 } and I S { 0 , 1 } , 2 o p t for some other classes of graphs.