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Edge-maximal graphs of branchwidth $$k$$: The $$k$$-branches. (English) Zbl 1229.05153
Summary: Treewidth and branchwidth are two closely related connectivity parameters of graphs. Graphs of treewidth at most $$k$$ have well-known alternative characterizations as subgraphs of chordal graphs and as partial $$k$$-trees. In this paper we give analogous alternative characterizations for graphs of branchwidth at most $$k$$. We first show that they are the subgraphs of chordal graphs where every maximal clique $$X$$ has three subsets of size at most $$k$$ each such that any two subsets have union $$X$$, with the property that every minimal separator contained in $$X$$ is contained in one of the three subsets. Secondly, we give a characterization of the edge-maximal graphs of branchwidth $$k$$, that we call $$k$$-branches.

##### MSC:
 05C35 Extremal problems in graph theory 05C40 Connectivity 05C05 Trees
##### Keywords:
graph decomposition; chordal; branchwidth; edge maximal
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##### References:
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