On the Tractability of Covering a Graph with 2-Clubs

On the Parameterized Complexity of the $s$-Club Cluster Edge Deletion Problem

“…With such records at hand, we then apply a standard DP algorithm over a tree decomposition of the input graph, which still requires a nontrivial amount of technicalities in order to update the records. Our records have some similarities, but also several key differences and extensions, with those used in a technique presented by Dondi and Lafond in [11,Thm. 14] to solve a different problem.…”

“…We are now ready to describe the items of the record to be stored at each bag. Some of these items are similar to those described in [11], although with some important differences that allow us to deal with any fixed value of s.…”

“…The third (and last) item of the record represents the key insight to extend the result in [11]. Roughly speaking, in the case s = 2, pairs of vertices in int(C) must have distance at most two in G[C], otherwise there is no C ⊇ C such that C is a 2-club of G. Unfortunately, this is not true in general when s > 2.…”

“…We are now ready to describe our DP algorithm over (X , T ). The main differences with respect to the algorithm in [11] lie in the management of the sets Q(•) (which do not exist in [11]), and on a more sophisticated updating of the tables D(•) and H(•), as a consequence of the non applicability of some simplifying assumptions that hold only when s = 2. Moreover, we also keep track of the number of edges having their end-vertices in different potential s-clubs.…”

“…Namely, they describe a fixed-parameter tractable algorithm for the problem of deciding whether the vertices of a graph can be covered with at most d (possibly overlapping) 2-clubs, parameterized by treewidth. The main novelties of the presented approach with respect to [11] will be suitably highlighted throughout the paper.…”