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Dense trees: a new look at degenerate graphs. (English) Zbl 1103.05085
A \(k\)-dense forest (tree) is a (connected) graph of which each subgraph contains a vertex of degree at most \(k\). That is, the \(1\)-dense trees are exactly the trees, and all partial \(k\)-trees are \(k\)-dense forests, see H. L. Bodlaender [Theor. Comput. Sci. 209, 1–45 (1998; Zbl 0912.68148)]. \(k\)-dense trees were studied in connection with first-fit colouring by G. Szekeres and H. S. Wilf [J. Comb. Theory 4, 1–3 (1967; Zbl 0171.44901)] and D. W. Matula [SIAM Rev. 10, 481–482 (1968)], and also by D. R. Lick and A. T. White [Can. J. Math. 22, 1082–1096 (1970; Zbl 0202.23502)], who called them \(k\)-degenerate graphs, and by E. C. Freuder [J. Assoc. Comput. Mach. 29, 24–32 (1982; Zbl 0477.68063)]. Here the focus is on maximal \(k\)-dense trees, their relation to \(k\)-trees, and the decomposition of \(k\)-dense trees into \(k\)-spanning trees.
MSC:
05C85 Graph algorithms (graph-theoretic aspects)
05C05 Trees
05C35 Extremal problems in graph theory
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