Deformable Polygon Representation and Near-Mincuts

We derive a necessary and sufficient condition for a symmetric family of sets to have a geometric representation involving a convex polygon and some of its diagonals. We show that cuts of value less than 6/5 times the edge-connectivity of a graph admit such a representation, thereby extending the cactus representation of all mincuts.

[1]  Ludwig Schl fli.  Theorie der vielfachen Kontinuit??t , 1950 .

[2]  A. Benczúr.  The structure of near-minimum edge cuts , 1994 .

[3]  Myriam Preissmann,et al.  Graphs with Largest Number of Minimum Cuts , 1996, Discret. Appl. Math..

[4]  András A. Benczúr,et al.  Cut structures and randomized algorithms in edge-connectivity problems , 1997 .

[5]  Michel X. Goemans,et al.  Minimizing submodular functions over families of sets , 1995, Comb..

[6]  L. Schläfli.  Theorie der vielfachen Kontinuität , 1901 .

[7]  David P. Williamson,et al.  On the Number of Small Cuts in a Graph , 1996, Inf. Process. Lett..

[8]  Hiroshi Nagamochi,et al.  Canonical cactus representation for minimum cuts , 1994 .

[9]  Vijay V. Vazirani,et al.  Representing and Enumerating Edge Connectivity Cuts in RNC , 1991, WADS.

[10]  Hiroshi Nagamochi.  Constructing a cactus for minimum cust og a graph in O(mn+n2log n) time and O(m) space , 2003 .

[11]  Zeev Nutov,et al.  A 2-level cactus model for the system of minimum and minimum+1 edge-cuts in a graph and its incremental maintenance , 1995, STOC '95.

[12]  Jiri Matousek,et al.  Lectures on discrete geometry , 2002, Graduate texts in mathematics.

[13]  Lisa Fleischer.  Building Chain and Cactus Representations of All Minimum Cuts from Hao-Orlin in the Same Asymptotic Run Time , 1998, IPCO.

[14]  Kyoto Unzverszty,et al.  CONSTRUCTING CACTUS REPRESENTATION FOR ALL MINIMUM CUTS IN AN UNDIRECTED NETWORK , 2004 .

[15]  A. Tucker,et al.  A structure theorem for the consecutive 1's property☆ , 1972 .

[16]  Toshihide Ibaraki,et al.  Computing All Small Cuts in an Undirected Network , 1997, SIAM J. Discret. Math..

[17]  András A. Benczúr,et al.  A representation of cuts within 6/5 times the edge connectivity with applications , 1995, Proceedings of IEEE 36th Annual Foundations of Computer Science.

[18]  E. A. Timofeev,et al.  Efficient algorithm for finding all minimal edge cuts of a nonoriented graph , 1986 .

[19]  Toshihide Ibaraki,et al.  A fast algorithm for cactus representations of minimum cuts , 2000 .

[20]  Toshihide Ibaraki,et al.  Computing All Small Cuts in Undirected Networks , 1994, ISAAC.

[21]  Harold N. Gabow,et al.  Applications of a poset representation to edge connectivity and graph rigidity , 1991, [1991] Proceedings 32nd Annual Symposium of Foundations of Computer Science.

[22]  P. Duchet.  Classical Perfect Graphs: An introduction with emphasis on triangulated and interval graphs , 1984 .

[23]  L. Lovász.  Combinatorial problems and exercises , 1979 .

[24]  Alain Quilliot.  Circular representation problem on hypergraphs , 1984, Discret. Math..