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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
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