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.
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Song, Y., Liu, T., Xu, K. (2012). Independent Domination on Tree Convex Bipartite Graphs. In: Snoeyink, J., Lu, P., Su, K., Wang, L. (eds) Frontiers in Algorithmics and Algorithmic Aspects in Information and Management. Lecture Notes in Computer Science, vol 7285. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-29700-7_12
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DOI: https://doi.org/10.1007/978-3-642-29700-7_12
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