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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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##### References:
 [1] Arbib, C.; Flammini, M.; Nardelli, E., How to survive while visiting a graph, Discrete appl. math., 99, 279-293, (2000) · Zbl 0940.05062 [2] Bodlaender, H.L., A partial k-arboretum of graphs with bounded treewidth, Theoret. comput. sci., 209, 1-45, (1998) · Zbl 0912.68148 [3] Diestel, R., Graph theory, (2000), Springer-Verlag New York · Zbl 0945.05002 [4] F. Flores Pacheco, G. Franceschini, F. Luccio, L. Pagli, Decomposition of k-dense, in: Proc. Distributed Data and Structures 3, Carleton Scientific, Ottawa, 2001, pp. 11-23 [5] G. Franceschini, Dense tree: una generalizzazione del concetto di albero, Master’s Thesis, University of Pisa (2000) [6] Freuder, E.C., A sufficient condition for backtrack-free search, J. ACM, 29, 24-32, (1982) · Zbl 0477.68063 [7] Garey, M.R.; Johnson, D.S., Computers and intractability, (1979), Freeman San Francisco, CA · Zbl 0411.68039 [8] Kloks, T., Treewidth. computations and approximations, Lecture notes in computer science, vol. 842, (1994), Springer-Verlag Berlin · Zbl 0825.68144 [9] Lick, D.; White, A., k-degenerate graphs, Can. J. math., 22, 1082-1096, (1970) · Zbl 0202.23502 [10] F. Luccio, A. Mesa, L. Pagli, A distributed tree data structure, in: Proc. 1st Internat. Symp. on Advanced Distributed Systems (ISADS2000), Guadalajara, Mexico, 2000, pp. 1-6 [11] F. Luccio, L. Pagli, Dense trees: a new structure for interconnection, in: Proc. Distributed Data and Structures 2, Carleton Scientific, Ottawa, 2000, pp. 56-72 [12] Matula, D.W., A min – max theorem for graphs with applications to graph coloring, SIAM rev., 10, 481-482, (1968) [13] Nardelli, E.; Proietti, G.; Widmaer, P., Swapping a failing edge of a single source shortest path tree is good and fast, Algorithmica, 35, 56-74, (2003) · Zbl 1026.68102 [14] Peterson, L.L.; Davie, B.S., Computer networks: A systems approach, (2003), Morgan Kaufmann San Francisco, CA [15] Szekeres, G.; Wilf, H.S., An inequality for the chromatic number of a graph, J. combin. theory, 4, 1-3, (1968) · Zbl 0171.44901
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