Imbalance, Cutwidth, and the Structure of Optimal Orderings

Imbalance Parameterized by Twin Cover Revisited

“…The second result is based on the ILP formulation that we alluded to earlier. We mention here that we rely crucially on the structural result of [7] for arguing the correctness of our algorithmic approaches.…”

“…Further, the problem is known to be FPT when parameterized by imbalance [11], vertex cover [5], neighborhood diversity [1], and the combined parameter treewidth and maximum degree [11]. Recently, it was claimed that imbalance is also FPT when parameterized by twin cover [7], which is a substantial improvement over the vertex cover parameter. We mention briefly here that a vertex cover of a graph G is a subset S ⊆ V(G) such that G \ S is an independent set, and a twin cover of a graph is a subset T ⊆ V(G) such that the connected components of G \ S consist of vertices which are true twins -in particular, note that each connected component induces a clique, and further, all vertices have the same neighborhood in the cover T .…”

“…We mention briefly here that a vertex cover of a graph G is a subset S ⊆ V(G) such that G \ S is an independent set, and a twin cover of a graph is a subset T ⊆ V(G) such that the connected components of G \ S consist of vertices which are true twins -in particular, note that each connected component induces a clique, and further, all vertices have the same neighborhood in the cover T . The method employed in [7] to obtain a FPT algorithm for Imbalance parameterized by twin cover relies on a structural lemma which, roughly speaking, states that there exist optimal orderings where any maximal collection of true twins appear together. Based on this, it is claimed that the cliques of G \ S can be contracted into singleton vertices to obtain an equivalent instance H. By observing that the twin cover of G is a vertex cover of H, we may now use the FPT algorithm for Imbalance parameterized by vertex cover to obtain an imbalance-optimal ordering for H, and the contracted vertices can be "expanded back in place" to recover an optimal ordering for G.…”

Characterization of the Imbalance Problem on Complete Bipartite Graphs

“…The imbalance problem is NP-complete for several graph classes, including bipartite graphs with degree at most 6, weighted trees [1], general graphs with degree at most 4 [6], and split graphs [4]. The problem becomes polynomial time solvable on superfragile graphs [4]. The problem is linear time solvable on proper interval graphs [4], bipartite permutation graphs, and threshold graphs [3].…”

“…The problem becomes polynomial time solvable on superfragile graphs [4]. The problem is linear time solvable on proper interval graphs [4], bipartite permutation graphs, and threshold graphs [3].…”