Independent domination on tree convex bipartite graphs

  • Authors:

  • Yu Song;Tian Liu;Ke Xu

  • Affiliations:

  • Key Laboratory of High Confidence Software Technologies, Ministry of Education, Institute of Software, School of Electronic Engineering and Computer Science, Peking University, Beijing, China;Key Laboratory of High Confidence Software Technologies, Ministry of Education, Institute of Software, School of Electronic Engineering and Computer Science, Peking University, Beijing, China;National Lab. of Software Development Environment, Beihang University, Beijing, China

  • Venue:
  • FAW-AAIM'12 Proceedings of the 6th international Frontiers in Algorithmics, and Proceedings of the 8th international conference on Algorithmic Aspects in Information and Management
  • Year:
  • 2012

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Abstract

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.