Independent Domination on Tree Convex Bipartite Graphs

  • Yu Song
  • Tian Liu
  • Ke Xu
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7285)


An independent dominating set in a graph is a subset of vertices, such that every vertex outside this subset has a neighbor in this subset (dominating), and the induced subgraph of this subset contains no edge (independent). It was known that finding the minimum independent dominating set (Independent Domination) is \(\cal{NP}\)-complete on bipartite graphs, but tractable on convex bipartite graphs. A bipartite graph is called tree convex, if there is a tree defined on one part of the vertices, such that for every vertex in another part, the neighborhood of this vertex is a connected subtree. A convex bipartite graph is just a tree convex one where the tree is a path. We find that the sum of larger-than-two degrees of the tree is a key quantity to classify the computational complexity of independent domination on tree convex bipartite graphs. That is, when the sum is bounded by a constant, the problem is tractable, but when the sum is unbounded, and even when the maximum degree of the tree is bounded, the problem is \(\cal{NP}\)-complete.


Bipartite Graph Maximum Degree Chordal Graph Minimum Cardinality Satisfying Assignment 
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  1. 1.
    Damaschke, P., Muller, H., Kratsch, D.: Domination in Convex and Chordal Bipartite Graphs Information Processing Letters 36, 231–236 (1990)Google Scholar
  2. 2.
    Farber, M.: Independent Domination in Chordal Graphs. Operations Research Letters 1, 134–138 (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Garey, M.R., Johnson, D.S.: Computers and Intractability. A Guide to the Theory of NP-Completeness (1979)Google Scholar
  4. 4.
    Irving, W.: On approximating the minimum independent dominating set. Information Processing Letters 37, 197–200 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Jiang, W., Liu, T., Ren, T.N., Xu, K.: Two Hardness Results on Feedback Vertex Sets. In: Atallah, M., Li, X.-Y., Zhu, B. (eds.) FAW-AAIM 2011. LNCS, vol. 6681, pp. 233–243. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  6. 6.
    Jiang, W., Liu, T., Xu, K.: Tractable Feedback Vertex Sets in Restricted Bipartite Graphs. In: Wang, W., Zhu, X., Du, D.-Z. (eds.) COCOA 2011. LNCS, vol. 6831, pp. 424–434. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  7. 7.
    Muller, H., Brandstadt, A.: The NP-completeness of Steiner Tree and Dominating Set for Chordal Bipartite Graphs. Theoretical Computer Science 53, 257–265 (1987)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Yu Song
    • 1
  • Tian Liu
    • 1
  • Ke Xu
    • 2
  1. 1.Key Laboratory of High Confidence Software Technologies, Ministry of Education, Institute of Software, School of Electronic Engineering and Computer SciencePeking UniversityBeijingChina
  2. 2.National Lab. of Software Development EnvironmentBeihang UniversityBeijingChina

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