, But can it be approximated? Moreover, we showed that those parameters can be computed in polynomial (and even linear) time for trees. It would be then natural to identify other classes of graphs for which this can also be done. We then have the following questions. Problem 6.4. For which classes of graphs, can we ? compute BOMD (resp. BOMD * ) in polynomial time? ? approximate BOMD (resp. BOMD * ) in polynomial time? ? decide whether BOMD = BOMD * in polynomial time? Regarding the complexity of deciding whether BOMD(G) ? k (and similarly BOMD * (G) ? k) for a given graph G, we have proved in Corollary 3.6 that it is NP-hard for every k ? 1. Looking at our proof, however, it can be noted that this result only holds for G being non-connected. Since we mostly focused on connected graphs throughout this work, minimum functions f and f * such that |V

. Womd-*, G)) is at most k, we do not know the complexity of computing WOMD(G) (resp

, But what about other classes of graphs? Hence we are left with the analogous questions to those which classes of graphs, can we ? compute WOMD (resp. WOMD * ) in polynomial time?, ? approximate WOMD (resp. WOMD * ) in polynomial time? ? decide whether WOMD = WOMD * in polynomial time

, In addition, given a tree T , one can compute WOMD(T ) in linear time. This suggests that there might be a nice characterisation of ?-trees (and of (? ? 1)-trees at the same time), i.e. a simple description of what the precise structure of ?-trees is, Finally, recall that there are two kinds of trees T with respect to WOMD: (? ? 1)-trees T for which WOMD(T ) = ?(T ) ? 1, and ?-trees T for which WOMD(T ) = ?(T )

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