Independence and Efficient Domination on P₆-free Graphs

Abstract

In the Maximum Weight Independent Set problem, the input is a graph G, every vertex has a nonnegative integer weight, and the task is to find a set S of pairwise non-adjacent vertices, maximizing the total weight of the vertices in S. We give an n O(log 2 n)time algorithm for this problem on graphs excluding the path P 6 on 6 vertices as an induced subgraph. Currently, there is no constant k known for which Maximum Weight Independent Set on P k -free graphs becomes NP-complete, and our result implies that if such a k exists, then k > 6 unless all problems in NP can be decided in (quasi)polynomial time.Using the combinatorial tools that we develop for the above algorithm, we also give a polynomial-time algorithm for Maximum Weight Efficient Dominating Set on P 6 -free graphs. In this problem, the input is a graph G, every vertex has an integer weight, and the objective is to find a set S of maximum weight such that every vertex in G has exactly one vertex in S in its closed neighborhood, or to determine that no such set exists. Prior to our work, the class of P 6 -free graphs was the only class of graphs defined by a single forbidden induced subgraph on which the computational complexity of Maximum Weight Efficient Dominating Set was unknown.

Maximum Weight Independent Sets for (P7,triangle)-free graphs in polynomial time

“…Recently, Lokshtanov et al [26] proved that MWIS can be solved for P 5 -free graphs in polynomial time, and Lokshtanov et al [25] proved that MWIS can be solved for P 6 -free graphs in quasi-polynomial time n O(log 2 n) . A natural open problem is whether MWIS can be solved for P k -free graphs in polynomial time for k ≥ 6 or in quasipolynomial time for k ≥ 7 − even though some characterizations of P k -free graphs and some progress are known in the literature [13].…”

“…However Lokshtanov et al [25] recently proved that MWIS can be solved for P 6 -free graphs in quasi-polynomial time n O(log 2 n) . Then − also referring to the conclusions of [26] − a natural open problem is to establish if MWIS can be solved for P k -free graphs in polynomial time for k ≥ 6 or in quasipolynomial time for k ≥ 7, even though some characterizations of P k -free graphs and some progress are known in the literature; see e.g.…”

“…Its complexity was an open problem for P k -free graphs, k ≥ 5. Recently, Lokshtanov et al [26] proved that MWIS can be solved in polynomial time for P 5 -free graphs, and Lokshtanov et al [25] proved that MWIS can be solved in quasi-polynomial time for P 6 -free graphs. It still remains an open problem whether MWIS can be solved in polynomial time for P k -free graphs, k ≥ 6 or in quasi-polynomial time for P k -free graphs, k ≥ 7.…”

Excluding Hooks and their Complements

“…The work for P 5 -free graphs [32] follows the same approach, but generalises it, by showing that in P 5 -free graphs one needs to examine only a particular (polynomially-sized) set of potential maximal cliques in the aforementioned algorithm. Subsequent work [31] uses minimal separators and potential maximal cliques in a different way to develop a quasipolynomial-time algorithm for the Independent Set problem in P 6 -free graphs.…”

Weighted efficient domination in two subclasses of P6-free graphs

“…Very recently, Lokshtanov, Pilipczuk and van Leeuwen [24] and independently, Mosca [31] showed that ED is solvable in polynomial time for P 6 -free graphs.…”