On the complexity of dominating set problems related to the minimum all-ones problem

  • Authors:

  • Hajo Broersma;Xueliang Li

  • Affiliations:

  • Department of Computer Science, Durham University, Science Labs, South Street, DH1 3LE, Durham, United Kingdom and Center for Combinatorics and LPMC, Nankai University, Tianjin 300071, Peoples Rep ...;Center for Combinatorics and LPMC, Nankai University, Tianjin 300071, Peoples Republic of China

  • Venue:
  • Theoretical Computer Science
  • Year:
  • 2007

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Abstract

The minimum all-ones problem and the connected odd dominating set problem were shown to be NP-complete in different papers for general graphs, while they are solvable in linear time (or trivial) for trees, unicyclic graphs, and series-parallel graphs. The complexity of both problems when restricted to bipartite graphs was raised as an open question. Here we solve both problems. For this purpose, we introduce the related decision problem of the existence of an odd dominating set without isolated vertices, and study its complexity. Our main result shows that this new problem is NP-complete, even when restricted to bipartite graphs. We use this result to deduce that the minimum all-ones problem and the connected odd dominating set problem are also NP-complete for bipartite graphs. We show that all three problems are solvable in linear time for graphs with bounded treewidth. We also show that the new problem remains NP-complete when restricted to other graph classes, e.g., planar graphs, graphs with girth at least five, and graphs with a small maximum degree, in particular 3-regular graphs.