Bounds on isolated scattering number

The isolated scattering number is a parameter that measures the vulnerability of networks. This measure is bounded by formulas depending on the independence number. We present new bounds on the isolated scattering number that can be calculated in polynomial time. References Z. Chen, M. Dehmer, F. Emmert-Streib, and Y. Shi. Modern and interdisciplinary problems in network science: A translational research perspective. CRC Press, 2018. doi: 10.1201/9781351237307 P. Erdős and T. Gallai. On the minimal number of vertices representing the edges of a graph. Magyar Tud. Akad. Mat. Kutató Int. Közl. 6 (1961), pp. 181–203. url: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.210.7468 J. Harant and I. Schiermeyer. On the independence number of a graph in terms of order and size. Discrete Math. 232.1–3 (2001), pp. 131–138. doi: 10.1016/S0012-365X(00)00298-3 E. Korach, T. Nguyen, and B. Peis. Subgraph characterization of red/blue-split graph and Kőnig Egerváry graphs. Proceedings of the Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithms. ACM, New York, 2006, pp. 842–850. doi: 10.1145/1109557.1109650 F. Li, Q. Ye, and Y. Sun. Proceedings of the 2016 Joint Conference of ANZIAM and Zhejiang Provincial Applied Mathematics Association, ANZPAMS-2016. Ed. by P. Broadbridge, M. Nelson, D. Wang, and A. J. Roberts. Vol. 58. ANZIAM J. 2017, E81–E97. doi: 10.21914/anziamj.v58i0.10993 F. Li, Q. Ye, and X. Zhang. Isolated scattering number of split graphs and graph products. ANZIAM J. 58.3-4 (2017), pp. 350–358. doi: 10.1017/S1446181117000062 E. R. Scheinerman and D. H. Ullman. Fractional graph theory. Dover Publications, 2011. url: https://www.ams.jhu.edu/ers/wp-content/uploads/2015/12/fgt.pdf S. Y. Wang, Y. X. Yang, S. W. Lin, J. Li, and Z. M. Hu. The isolated scattering number of graphs. Acta Math. Sinica (Chin. Ser.) 54.5 (2011), pp. 861–874. url: http://www.actamath.com/EN/abstract/abstract21097.shtml M. Xiao and H. Nagamochi. Exact algorithms for maximum independent set. Inform. and Comput. 255, Part 1 (2017), pp. 126–146. doi: 10.1016/j.ic.2017.06.001

Smaller Cuts, Higher Lower Bounds

This article proves strong lower bounds for distributed computing in the congest model, by presenting the bit-gadget : a new technique for constructing graphs with small cuts. The contribution of bit-gadgets is twofold. First, developing careful sparse graph constructions with small cuts extends known techniques to show a near-linear lower bound for computing the diameter, a result previously known only for dense graphs. Moreover, the sparseness of the construction plays a crucial role in applying it to approximations of various distance computation problems, drastically improving over what can be obtained when using dense graphs. Second, small cuts are essential for proving super-linear lower bounds, none of which were known prior to this work. In fact, they allow us to show near-quadratic lower bounds for several problems, such as exact minimum vertex cover or maximum independent set, as well as for coloring a graph with its chromatic number. Such strong lower bounds are not limited to NP-hard problems, as given by two simple graph problems in P, which are shown to require a quadratic and near-quadratic number of rounds. All of the above are optimal up to logarithmic factors. In addition, in this context, the complexity of the all-pairs-shortest-paths problem is discussed. Finally, it is shown that graph constructions for congest lower bounds translate to lower bounds for the semi-streaming model, despite being very different in its nature.

THE STRUCTURE OF GRAPHS ON n VERTICES WITH THE DEGREE SUM OF ANY TWO NONADJACENT VERTICES EQUAL TO n-2

Let G be an undirected simple graph on n vertices and sigma2(G)=n-2 (degree sum of any two non-adjacent vertices in G is equal to n-2) and alpha(G) be the cardinality of an maximum independent set of G. In G, a vertex of degree (n-1) is called total vertex. We show that, for n>=3 is an odd number then alpha(G)=2 and G is a disconnected graph; for n>=4 is an even number then 2=<alpha(G)<=(n+2)/2, where if alpha(G)=2 then G is a disconnected graph, otherwise G is a connected graph, G contains k total vertices and n-k vertices of degree delta=(n-2)/2, where 0<=k<=(n-2)/2. In particular, when k=0 then G is an (n-2)/2-Regular graph.