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. n 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 \(q \not\in \{3,5\}\), 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.
This extended abstract does not contain all proofs. For a full version, please refer to [14] or contact one of the authors.
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Paul, C., Proskurowski, A., Telle, J.A. (2006). Generation of Graphs with Bounded Branchwidth. In: Fomin, F.V. (eds) Graph-Theoretic Concepts in Computer Science. WG 2006. Lecture Notes in Computer Science, vol 4271. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11917496_19
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DOI: https://doi.org/10.1007/11917496_19
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