Counting Independent Sets of a Fixed Size in Graphs with Given Minimum Degree

Galvin showed that for all fixed δ and sufficiently large n, the n-vertex graph with minimum degree δ that admits the most independent sets is the complete bipartite graph . He conjectured that except perhaps for some small values of t, the same graph yields the maximum count of independent sets of size t for each possible t. Evidence for this conjecture was recently provided by Alexander, Cutler, and Mink, who showed that for all triples with , no n-vertex bipartite graph with minimum degree δ admits more independent sets of size t than . Here, we make further progress. We show that for all triples with and , no n-vertex graph with minimum degree δ admits more independent sets of size t than , and we obtain the same conclusion for and . Our proofs lead us naturally to the study of an interesting family of critical graphs, namely those of minimum degree δ whose minimum degree drops on deletion of an edge or a vertex

Preferential survival in models of complex ad hoc networks

There has been a rich interplay in recent years between (i) empirical investigations of real world dynamic networks, (ii) analytical modeling of the microscopic mechanisms that drive the emergence of such networks, and (iii) harnessing of these mechanisms to either manipulate existing networks, or engineer new networks for specific tasks. We continue in this vein, and study the deletion phenomenon in the web by following two different sets of web-sites (each comprising more than 150,000 pages) over a one-year period. Empirical data show that there is a significant deletion component in the underlying web networks, but the deletion process is not uniform. This motivates us to introduce a new mechanism of preferential survival (PS), where nodes are removed according to a degree-dependent deletion kernel. We use the mean-field rate equation approach to study a general dynamic model driven by Preferential Attachment (PA), Double PA (DPA), and a tunable PS, where c nodes (c<1) are deleted per node added to the network, and verify our predictions via large-scale simulations. One of our results shows that, unlike in the case of uniform deletion, the PS kernel when coupled with the standard PA mechanism, can lead to heavy-tailed power law networks even in the presence of extreme turnover in the network. Moreover, a weak DPA mechanism, coupled with PS, can help make the network even more heavy-tailed, especially in the limit when deletion and insertion rates are almost equal, and the overall network growth is minimal. The dynamics reported in this work can be used to design and engineer stable ad hoc networks and explain the stability of the power law exponents observed in real-world networks.Comment: 9 pages, 6 figure

Counterpart semantics for a second-order mu-calculus

We propose a novel approach to the semantics of quantified μ-calculi, considering models where states are algebras; the evolution relation is given by a counterpart relation (a family of partial homomorphisms), allowing for the creation, deletion, and merging of components; and formulas are interpreted over sets of state assignments (families of substitutions, associating formula variables to state components). Our proposal avoids the limitations of existing approaches, usually enforcing restrictions of the evolution relation: the resulting semantics is a streamlined and intuitively appealing one, yet it is general enough to cover most of the alternative proposals we are aware of

Tree-edges deletion problems with bounded diameter obstruction sets

AbstractWe study the following problem: given a tree G and a finite set of trees H, find a subset O of the edges of G such that G-O does not contain a subtree isomorphic to a tree from H, and O has minimum cardinality. We give sharp boundaries on the tractability of this problem: the problem is polynomial when all the trees in H have diameter at most 5, while it is NP-hard when all the trees in H have diameter at most 6. We also show that the problem is polynomial when every tree in H has at most one vertex with degree more than 2, while it is NP-hard when the trees in H can have two such vertices.The polynomial-time algorithms use a variation of a known technique for solving graph problems. While the standard technique is based on defining an equivalence relation on graphs, we define a quasiorder. This new variation might be useful for giving more efficient algorithm for other graph problems