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Algorithmic Generation of Graphs of Branch-width ≤ k

Abstract : Branchwidth is a connectivity parameter of graphs closely related to treewidth. Graphs of treewidth at most k can be generated algorithmically as the subgraphs of k-trees. In this paper, we investigate the family of edge-maximal graphs of branchwidth k, that we call k-branches. The k-branches are, just as the k-trees, a subclass of the chordal graphs where all minimal separators have size k. However, a striking difference arises when considering subgraph-minimal members of the family. Whereas K_{k+1} is the only subgraph-minimal k-tree, we show that for any k ≤ 7 a minimal k-branch having q maximal cliques exists for any value of q different than 3 and 5, except for k=8,q=2. We characterize subgraph-minimal k-branches for all values of k. Our investigation leads to a generation algorithm, that adds one or two new maximal cliques in each step, producing exactly the k-branches.
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Contributor : Christophe Paul <>
Submitted on : Thursday, September 25, 2008 - 1:53:03 PM
Last modification on : Friday, November 20, 2020 - 4:22:03 PM


  • HAL Id : lirmm-00324551, version 1



Christophe Paul, Andrzej Proskurowski, Jan Arne Telle. Algorithmic Generation of Graphs of Branch-width ≤ k. WG'06: Graph Theoretical Concepts in Computer Science, pp.206-216. ⟨lirmm-00324551⟩



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