A subset of vertices in a graph is called a dissociation set if it induces a subgraph with vertex degree at most 1. A dissociation set D is maximal if no other dissociation set contains D. The complexity of finding a dissociation set of maximum size in line graphs and finding a maximal dissociation set of minimum size in general graphs is considered.

The minimum dominating set (MDS) problem is a fundamental subject of theoretical computer science, and has found vast applications in different areas, including sensor networks, protein interaction networks, and structural controllability. However, the determination of the size of a MDS and the number of all MDSs in a general network is NP-hard, and it thus makes sense to seek particular graphs...

We show that the structure R of recursively enumerable degrees is not a Σ1-elementary substructure of Dn, where Dn (n > 1) is the structure of n-r.e. degrees in Ershov hierarchy.

An ordered biclique partition of the complete graph Kn on n vertices is a collection of bicliques (i.e., complete bipartite graphs) such that (i) every edge of Kn is covered by at least one and at most two bicliques in the collection, and (ii) if an edge e is covered by two bicliques then each endpoint of e is in the first class in one of these bicliques and in the second class in other one. We...

We survey some of the recent results on the structure of the computably enumerable (c.e.) sets under inclusion. Our main interest is on collections of c.e. sets which are closed under automorphic images, such as the orbit of a c.e. set, and their (Turing) degree theoretic and dynamic properties. We take an algebraic viewpoint rather than the traditional dynamic viewpoint.

There are several approaches addressing the problem of understanding the mechanism of color confinement in non-Abelian gauge theories. The most popular share the idea that only a subset of the degrees of freedom are relevant for confinement. In the Abelian projection approach, one takes into account the maximal Abelian subgroup U(1) of the gauge group SU(N) and monopoles are the effective degre...

We show that there are 5 formulas in the language of the Turing degrees, D, with ,_ and ^, that de…ne the relations x00 y00, x00 = y00 and so x 2 L2(y) = fx yjx00 = y00g in any jump ideal containing 0(!). There are also 6& 6 and 8 formulas that de…ne the relations w = x00 and w = x0, respectively, in any such ideal I. In the language with just the quanti…er complexity of each of these de…nition...

We prove that a K4-free graph G of order n, size m and maximum degree at most three has an independent set of cardinality at least 1 7 (4n−m− λ− tr) where λ counts the number of components of G whose blocks are each either isomorphic to one of four specific graphs or edges between two of these four specific graphs and tr is the maximum number of vertex-disjoint triangles in G. Our result genera...

A cluster of cycles (or (r; q)-polycycle) is a simple planar 2{connected nite or countable graph G of girth r and maximal vertex-degree q, which admits an (r; q)-polycyclic realization P(G) on the plane. An (r; q)-polycyclic realization is determined by the following properties: (i) all interior vertices are of degree q, (ii) all interior faces (denote their number by p r) are combinatorial r-g...

This is a non-standard paper, containing some problems in set theory I have in various degrees been interested in. Sometimes with a discussion on what I have to say; sometimes, of what makes them interesting to me, sometimes the problems are presented with a discussion of how I have tried to solve them, and sometimes with failed tries, anecdote and opinion. So the discussion is quite personal, ...