Abstract
Given an undirected 3-connected graph G with n vertices and m edges, we modify depth-first search to produce a sparse spanning subgraph with at most 4n − 10 edges that is still 3-connected. If G is 2-connected, to maintain 2-connectivity, the resulting graph will have at most 2n − 3 edges. The way depth-first search discards irrelevant edges illustrates the reason behind its ability to verify and certify biconnectivity [1,2,3] and triconnectivity [4,5] in linear time. Dealing with a sparser graph, after the first depth-first-search calls, makes the algorithms in [2,5] more efficient. We also give a characterization of separation pairs of a 2-connected graph in terms of the resulting sparse graph.
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Elmasry, A. (2010). Why Depth-First Search Efficiently Identifies Two and Three-Connected Graphs. In: Cheong, O., Chwa, KY., Park, K. (eds) Algorithms and Computation. ISAAC 2010. Lecture Notes in Computer Science, vol 6507. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-17514-5_32
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DOI: https://doi.org/10.1007/978-3-642-17514-5_32
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