Every connected graph G with radius r(G) and independence number α(G) obeys α(G) ≥ r(G). Recently the graphs for which equality holds have been classified. Here we investigate the members of this class that are Cartesian products. We show that for non-trivial graphs G and H , α(G2H) = r(G2H) if and only if one factor is a complete graph on two vertices, and the other is a nontrivial complete gr...

Given an undirected graph G and a positive integer k, the k-vertex-connectivity augmentation problem is to find a smallest set F of new edges for which G + F is k-vertex-connected. Polynomial algorithms for this problem have been found only for k≤ 4 and a major open question in graph connectivity is whether this problem is solvable in polynomial time in general. In this paper we develop an algo...

Credal networks are graph-based statistical models whose parameters take values in a set, instead of being sharply specified as in traditional statistical models (e.g., Bayesian networks). The computational complexity of inferences on such models depends on the irrelevance/independence concept adopted. In this paper, we study inferential complexity under the concepts of epistemic irrelevance an...

Let ν be some graph parameter and let G be a class of graphs for which ν can be computed in polynomial time. In this situation it is often possible to devise a strategy to decide in polynomial time whether ν has a unique realization for some graph in G. We first give an informal description of the conditions that allow one to devise such a strategy, and then we demonstrate our approach for thre...

Action-Graph Games (AGGs) are a fully expressive game representation which can compactly express strict and context-specific independence and anonymity structure in players’ utility functions. We present an efficient algorithm for computing expected payoffs under mixed strategy profiles. This algorithm runs in time polynomial in the size of the AGG representation (which is itself polynomial in ...

Let G be a well covered graph, that is, all maximal independent sets of G have the same cardinality, and let ik denote the number of independent sets of cardinality k in G. We investigate the roots of the independence polynomial i(G, x)= ∑ ik xk . In particular, we show that if G is a well covered graph with independence number β, then all the roots of i(G, x) lie in in the disk |z| ≤β (this is...

Given two graphs G and H, assume that C = {C1, C2, . . . , Cq} is a clique cover of G and U is a subset of V (H). We introduce a new graph operation called the clique cover product, denoted by G ⋆ HU , as follows: for each clique Ci ∈ C , add a copy of the graph H and join every vertex of Ci to every vertex of U . We prove that the independence polynomial of G ⋆ HU I(G ⋆ H ;x) = I(H;x)I(G; xI(H...

In this paper we compare and illustrate the algorithmic use of graphs of bounded treewidth and graphs of bounded clique-width. For this purpose we give polynomial time algorithms for computing the four basic graph parameters independence number, clique number, chromatic number, and clique covering number on a given tree structure of graphs of bounded tree-width and graphs of bounded clique-widt...

We prove a lemma regarding the linear independence of certain vectors and use it to improve on a bound due to Schmidt on the zero-multiplicity of linear recurrence sequences. 1 Linear recurrence sequences Let {un}n∈Z be a linear recurrence sequence of complex numbers satisfying the recurrence relation un = c1un−1 + · · ·+ ctun−t, (1) for c1, . . . , ct ∈ C with ct 6= 0. We say it is of order t ...

We present a chosen-ciphertext attack against the public key cryptosystem called NTRU. This cryptosystem is based on polynomial algebra. Its security comes from the interaction of the polynomial mixing system with the independence of reduction modulo two relatively prime integers p and q. In this paper, we examine the effect of feeding special polynomials built from the public key to the decryp...