Computing the Degeneracy of Large Graphs

Any ordering of the nodes of an n-node, m-edge simple undirected graph G defines an acyclic orientation of the edges in which each edge is oriented from the earlier node in the ordering to the later. The degeneracy on an ordering is the maximum outdegree it induces, and the degeneracy of a graph is smallest degeneracy of any node ordering. Small-degeneracy orderings have many applications.

[1]  Harold N. Gabow,et al.  Forests, frames, and games: algorithms for matroid sums and applications , 1988, STOC '88.

[2]  Moses Charikar,et al.  Greedy approximation algorithms for finding dense components in a graph , 2000, APPROX.

[3]  Joan Feigenbaum,et al.  On graph problems in a semi-streaming model , 2005, Theor. Comput. Sci..

[4]  Christoph Lenzen,et al.  Minimum Dominating Set Approximation in Graphs of Bounded Arboricity , 2010, DISC.

[5]  Edward R. Scheinerman,et al.  On the thickness and arboricity of a graph , 1991, J. Comb. Theory, Ser. B.

[6]  Zdenek Dvorak,et al.  Constant-factor approximation of domination number in sparse graphs , 2011, ArXiv.

[7]  Graham Cormode,et al.  An improved data stream summary: the count-min sketch and its applications , 2004, J. Algorithms.

[8]  Lukasz Kowalik,et al.  Approximation Scheme for Lowest Outdegree Orientation and Graph Density Measures , 2006, ISAAC.

[9]  Ziv Bar-Yossef,et al.  Reductions in streaming algorithms, with an application to counting triangles in graphs , 2002, SODA '02.

[10]  Klaus Jansen,et al.  Experimental and Efficient Algorithms , 2003, Lecture Notes in Computer Science.

[11]  Da-Wei Wang,et al.  Efficient parallel I/O scheduling in the presence of data duplication , 2003, 2003 International Conference on Parallel Processing, 2003. Proceedings..

[12]  Leland L. Beck,et al.  Smallest-last ordering and clustering and graph coloring algorithms , 1983, JACM.

[13]  Matthias Ruhl,et al.  Efficient algorithms for new computational models , 2003 .

[14]  Sudipto Guha,et al.  Linear programming in the semi-streaming model with application to the maximum matching problem , 2011, Inf. Comput..

[15]  Moses Charikar,et al.  Finding frequent items in data streams , 2004, Theor. Comput. Sci..

[16]  Robin Milner,et al.  On Observing Nondeterminism and Concurrency , 1980, ICALP.

[17]  Marios Hadjieleftheriou,et al.  Finding frequent items in data streams , 2008, Proc. VLDB Endow..

[18]  Dorothea Wagner,et al.  Finding, Counting and Listing All Triangles in Large Graphs, an Experimental Study , 2005, WEA.

[19]  Jian Zhang,et al.  A Survey on Streaming Algorithms for Massive Graphs , 2010, Managing and Mining Graph Data.

[20]  B. Bollobás,et al.  Extremal Graph Theory , 2013 .

[21]  Noga Alon,et al.  Finding and counting given length cycles , 1997, Algorithmica.

[22]  Graham Cormode,et al.  Annotations in Data Streams , 2009, ICALP.

[23]  S. Muthukrishnan,et al.  Data streams: algorithms and applications , 2005, SODA '03.

[24]  Thomas C. O'Connell,et al.  A Survey of Graph Algorithms Under Extended Streaming Models of Computation , 2013, Fundamental Problems in Computing.

[25]  Anthony Mansfield,et al.  Determining the thickness of graphs is NP-hard , 1983, Mathematical Proceedings of the Cambridge Philosophical Society.

[26]  Koichi Yamazaki,et al.  Worst case analysis of a greedy algorithm for graph thickness , 2003, Inf. Process. Lett..

[27]  Andrew V. Goldberg,et al.  Finding a Maximum Density Subgraph , 1984 .

[28]  Klaus Jansen,et al.  Approximation Algorithms for Combinatorial Optimization , 2000 .