On the probe problem for (r,ℓ)-well-coveredness: Algorithms and complexity

Abstract

Let C be a class of graphs. A graph G=(V,E) is C-probe if V(G) can be partitioned into two sets: non-probesN and probesP, where N is an independent set and new edges may be added between some non-probe vertices such that the resulting graph is in the class C. In this case, we say that (N,P) is a C-probe partition of G. In the Unpartitioned Probe problem for a graph class C we are given a graph G and asked whether G has a C-probe partition, i.e., such a problem consist of recognizing the class of C-probe graphs. A graph G=(V,E) is an (r,ℓ)-graph when V can be partitioned into (S1,S2,…,Sr,K1,K2,…,Kℓ) such that S1,S2,…,Sr are independent sets, and K1,K2,…,Kℓ are cliques. A graph G is well-covered if every maximal independent set is also maximum, and it is (r,ℓ)-well-covered if it is well-covered as well as an (r,ℓ)-graph. In this paper, we study the complexity of the Unpartitioned Probe problem for the class of (r,ℓ)-well-covered graphs. We classify all but the (2,0) and (1,2) cases.