Compact group actions on equivariant absolute neighborhood retracts and their orbit spaces

We apply equivariant joins to give a new and more transparent proof of the following result: if G is a compact Hausdorff group and X a G-ANR (respectively, a G-AR), then for every closed normal subgroup H of G, the H-orbit space X/H is a G/H-ANR (respectively, a G/H-AR). In particular, X/G is an ANR (respectively, an AR).     

On the computation of the hull number of a graph

Let G be a graph. If u,v∈V(G), a u-vshortest path of G is a path linking u and v with minimum number of edges. The closed interval I[u,v] consists of all vertices lying in some u-v shortest path of G. For S⊆V(G), the set I[S] is the union of all sets I[u,v] for u,v∈S. We say that S is a convex set if I[S]=S. The convex hull of S, denoted I_h[S], is the smallest convex set containing S. A set S is a hull set of G if I_h[S]=V(G). The cardinality of a minimum hull set of G is the hull number of G, denoted by hn(G). In this work we prove that deciding whether hn(G)≤k is NP-complete.We also present polynomial-time algorithms for computing hn(G) when G is a unit interval graph, a cograph or a split graph.     

Groups of quasi-invariance and the Pontryagin duality

A Polish group G is called a group of quasi-invariance or a QI-group, if there exist a locally compact group X and a probability measure μ on X such that (1) there exists a continuous monomorphism ? from G into X with dense image, and (2) for each g∈X either g∈?(G) and the shift μg is equivalent to μ or g∉?(G) and μg is orthogonal to μ. It is proved that ?(G) is a σ-compact subset of X. We show that there exists a Polish non-locally quasi-convex (and hence nonreflexive) QI-group such that its bidual is not a QI-group. It is proved also that the bidual group of a QI-group may be not a saturated subgroup of X. It is constructed a reflexive non-discrete group topology on the integers.     

Formality theorem for Lie bialgebras and quantization of twists and coboundary r-matrices

Let (g,δ?) be a Lie bialgebra. Let (U?(g),Δ?) a quantization of (g,δ?) through Etingof-Kazhdan functor. We prove the existence of a L_∞-morphism between the Lie algebra C(g)=Λ(g) and the tensor algebra (without unit) T₊U=T₊(U?(g)[−1]) with Lie algebra structure given by the Gerstenhaber bracket. When s is a twist for (g,δ), we deduce from the formality morphism the existence of a quantum twist F. When (g,δ,r) is a coboundary Lie bialgebra, we get the existence of a quantization R of r.     

On derivable mappings

A linear mapping δ from an algebra A into an A-bimodule M is called derivable at c∈A if δ(a)b+aδ(b)=δ(c) for all a,b∈A with ab=c. For a norm-closed unital subalgebra A of operators on a Banach space X, we show that if C∈A has a right inverse in B(X) and the linear span of the range of rank-one operators in A is dense in X then the only derivable mappings at C from A into B(X) are derivations; in particular the result holds for all completely distributive subspace lattice algebras, J-subspace lattice algebras, and norm-closed unital standard algebras of B(X). As an application, every Jordan derivation from such an algebra into B(X) is a derivation. For a large class of reflexive algebras A on a Banach space X, we show that inner derivations from A into B(X) can be characterized by boundedness and derivability at any fixed C∈A, provided C has a right inverse in B(X). We also show that if A is a canonical subalgebra of an AF C^∗-algebra B and M is a unital Banach A-bimodule, then every bounded local derivation from A into M is a derivation; moreover, every bounded linear mapping from A into B that is derivable at the unit I is a derivation.     

Partitioning graphs into complete and empty graphs

Given integers j and k and a graph G, we consider partitions of the vertex set of G into j+k parts where j of these parts induce empty graphs and the remaining k induce cliques. If such a partition exists, we say G is a (j,k)-graph. For a fixed j and k we consider the maximum order n where every graph of order n is a (j,k)-graph. The split-chromatic number of G is the minimum j where G is a (j,j)-graph. Further, the cochromatic number is the minimum j+k where G is a (j,k)-graph. We examine some relations between cochromatic, split-chromatic and chromatic numbers. We also consider some computational questions related to chordal graphs and cographs.     

Selection over classes of ordinals expanded by monadic predicates

A monadic formula ψ(Y) is a selector for a monadic formula φ(Y) in a structure M if ψ defines in M a unique subset P of the domain and this P also satisfies φ in M. If C is a class of structures and φ is a selector for ψ in every M∈C, we say that φ is a selector for φ over C.For a monadic formula φ(X,Y) and ordinals α≤ω₁ and δ<ω^ω, we decide whether there exists a monadic formula ψ(X,Y) such that for every P⊆αof order-type smaller thanδ, ψ(P,Y) selects φ(P,Y) in (α,<). If so, we construct such a ψ.We introduce a criterion for a class C of ordinals to have the property that every monadic formula φ has a selector over it. We deduce the existence of S⊆ω^ω such that in the structure (ω^ω,<,S) every formula has a selector.Given a monadic sentence π and a monadic formula φ(Y), we decide whether φ has a selector over the class of countable ordinals satisfying π, and if so, construct one for it.     

On then-cutset property

For a partially ordered setP and an elementx ofP, a subsetS ofP is called a cutset forx inP if every element ofS is noncomparable tox and every maximal chain ofP meets {x}∪S. We letc(P) denote the smallest integerk such that every elementx ofP has a cutsetS with ‖S‖?k: Ifc(P)?n we say thatP has then-cutset property. Our results bear on the following question: givenP, what is the smallestn such thatP can be embedded in a partially ordered set having then-cutset property? As usual, 2 ⁿ denotes the Boolean lattice of all subsets of ann-element set, andB _n denotes the set of atoms and co-atoms of 2 ⁿ . We establish the following results: (i) a characterization, by means of forbidden configurations, of whichP can be embedded in a partially ordered set having the 1-cutset property; (ii) ifP contains a copy of 2 ⁿ , thenc(P)?2^[n/2]?1; (iii) for everyn>3 there is a partially ordered setP containing 2 ⁿ such thatc(P)<c(2 ⁿ ); (iv) for every positive integern there is a positive integerN such that, ifB _m is contained in a partially ordered set having then-cutset property, thenm?N.     

Building blocks for the variety of absolute retracts

Given a graph H with a labelled subgraph G, a retraction of H to G is a homomorphism r:H→G such that r(x)=x for all vertices x in G. We call G a retract of H. While deciding the existence of a retraction to a fixed graph G is NP-complete in general, necessary and sufficient conditions have been provided for certain classes of graphs in terms of holes, see for example Hell and Rival.For any integer k?2 we describe a collection of graphs that generate the variety AR_k of graphs G with the property that G is a retract of H whenever G is a subgraph of H and no hole in G of size at most k is filled by a vertex of H. We also prove that AR_k⊂NUF_k+1, where NUF_k+1 is the variety of graphs that admit a near unanimity function of arity k+1.     

On realizations of point determining graphs, and obstructions to full homomorphisms

A graph is point determining if distinct vertices have distinct neighbourhoods. A realization of a point determining graph H is a point determining graph G such that each vertex-removed subgraph G-x which is point determining, is isomorphic to H. We study the fine structure of point determining graphs, and conclude that every point determining graph has at most two realizations.A full homomorphism of a graph G to a graph H is a vertex mapping f such that for distinct vertices u and v of G, we have uv an edge of G if and only if f(u)f(v) is an edge of H. For a fixed graph H, a full H-colouring of G is a full homomorphism of G to H. A minimal H-obstruction is a graph G which does not admit a full H-colouring, such that each proper induced subgraph of G admits a full H-colouring. We analyse minimal H-obstructions using our results on point determining graphs. We connect the two problems by proving that if H has k vertices, then a graph with k+1 vertices is a minimal H-obstruction if and only if it is a realization of H. We conclude that every minimal H-obstruction has at most k+1 vertices, and there are at most two minimal H-obstructions with k+1 vertices.We also consider full homomorphisms to graphs H in which loops are allowed. If H has ? loops and k vertices without loops, then every minimal H-obstruction has at most (k+1)(?+1) vertices, and, when both k and ? are positive, there is at most one minimal H-obstruction with (k+1)(?+1) vertices.In particular, this yields a finite forbidden subgraph characterization of full H-colourability, for any graph H with loops allowed.     

Minimum cost homomorphisms to semicomplete multipartite digraphs

For digraphs D and H, a mapping f:V(D)→V(H) is a homomorphism ofDtoH if uv∈A(D) implies f(u)f(v)∈A(H). For a fixed directed or undirected graph H and an input graph D, the problem of verifying whether there exists a homomorphism of D to H has been studied in a large number of papers. We study an optimization version of this decision problem. Our optimization problem is motivated by a real-world problem in defence logistics and was introduced recently by the authors and M. Tso.Suppose we are given a pair of digraphs D,H and a cost c_i(u) for each u∈V(D) and i∈V(H). The cost of a homomorphism f of D to H is ∑_u∈V(D)c_f(u)(u). Let H be a fixed digraph. The minimum cost homomorphism problem for H, MinHOMP(H), is stated as follows: For input digraph D and costs c_i(u) for each u∈V(D) and i∈V(H), verify whether there is a homomorphism of D to H and, if it does exist, find such a homomorphism of minimum cost. In our previous paper we obtained a dichotomy classification of the time complexity of when H is a semicomplete digraph. In this paper we extend the classification to semicomplete k-partite digraphs, k≥3, and obtain such a classification for bipartite tournaments.     

Quasi-monotone mappings on θn-continua

A continuous function f from a continuum X onto a continuum Y is quasi-monotone if, for every subcontinuum M of Y with nonvoid interior, f^-1(M) has a finite number of components each of which is mapped onto M by f. A θ_n-continuum is one that no subcontinuum separates into more than n components. It is known that if f is quasi-monotone and X is a θ₁-continuum, then Y is a θ₁-continuum or a θ₂-continuum that is irreducible between two points. Examples are given to show that this cannot be generalized to a θ_n-continuum and n + 1 points for any n >1, but it is proved that if f is quasi-monotone and X is a θ_n-continuum, then Y is a θ_n-continuum or a θ_n+1-continuum that is the union of n + 2 continua H,S₁,S₂,…,S_n+1, whe for each i, S_i is the closure of a component of Y H, S_i is irreducible from some point P_i to H, and H is irreducible about its boundary. Some theorems and examples are given concerning the preservation of decomposition elements by a quasi-monotone map defined on a θ_n-continuum that admits a monotone, upper-semicontinuous decomposition onto a finite graph.     

Frobenius splitting and geometry of G-Schubert varieties

Let X be an equivariant embedding of a connected reductive group G over an algebraically closed field k of positive characteristic. Let B denote a Borel subgroup of G. A G-Schubert variety in X is a subvariety of the form diag(G)⋅V, where V is a B×B-orbit closure in X. In the case where X is the wonderful compactification of a group of adjoint type, the G-Schubert varieties are the closures of Lusztig's G-stable pieces. We prove that X admits a Frobenius splitting which is compatible with all G-Schubert varieties. Moreover, when X is smooth, projective and toroidal, then any G-Schubert variety in X admits a stable Frobenius splitting along an ample divisors. Although this indicates that G-Schubert varieties have nice singularities we present an example of a nonnormal G-Schubert variety in the wonderful compactification of a group of type G₂. Finally we also extend the Frobenius splitting results to the more general class of R-Schubert varieties.     

On the complexity of paths avoiding forbidden pairs

Given a graph G=(V,E), two fixed vertices s,t∈V and a set F of pairs of vertices (called forbidden pairs), the problem of a path avoiding forbidden pairs is to find a path from s to t that contains at most one vertex from each pair in F. The problem is known to be NP-complete in general and a few restricted versions of the problem are known to be in P. We study the complexity of the problem for directed acyclic graphs with respect to the structure of the forbidden pairs.We write x?y if and only if there exists a path from x to y and we assume, without loss of generality, that for every forbidden pair (x,y)∈F we have x?y. The forbidden pairs have a halving structure if no two pairs (u,v),(x,y)∈F satisfy v?x or v=x and they have a hierarchical structure if no two pairs (u,v),(x,y)∈F satisfy u?x?v?y. We show that the PAFP problem is NP-hard even if the forbidden pairs have the halving structure and we provide a surprisingly simple and efficient algorithm for the PAFP problem with the hierarchical structure.     

Bi-cycle extendable through a given set in balanced bipartite graphs

Let G=(X,Y;E) be a balanced bipartite graph of order 2n. The path-cover numberpc(H) of a graph H is the minimum number of vertex-disjoint paths that use up all the vertices of H. S⊆V(G) is called a balanced set of G if |S∩X|=|S∩Y|. In this paper, we will give some sufficient conditions for a balanced bipartite graph G satisfying that for every balanced set S, there is a bi-cycle of every length from |S|+2pc(〈S〉) up to 2n through S.     

A study of 3-arc graphs

An arc of a graph is an oriented edge and a 3-arc is a 4-tuple (v,u,x,y) of vertices such that both (v,u,x) and (u,x,y) are paths of length two. The 3-arc graph of a graph G is defined to have the arcs of G as vertices such that two arcs uv,xy are adjacent if and only if (v,u,x,y) is a 3-arc of G. In this paper, we study the independence, domination and chromatic numbers of 3-arc graphs and obtain sharp lower and upper bounds for them. We introduce a new notion of arc-coloring of a graph in studying vertex-colorings of 3-arc graphs.     

On iterates of fully closed absolutes

In this article we give a sufficient condition for a space X to have the fully closed absolute faX with the property fa(faX)=faX. An example of a compact space X such that the canonical mapping fa(α+1)X→fa(α)X (where α is a given ordinal) is not a homeomorphism is constructed. Also we give an example of a compact space X such that the canonical mapping faX→X is not a homeomorphism but for which there exists a homeomorphism faX→X.     

Characterizations of isomorphisms and derivations of some algebras

Let ? be a zero-product preserving bijective bounded linear map from a unital algebra A onto a unital algebra B such that ?(1)=k. We show that if A is a CSL algebra on a Hilbert space or a J-lattice algebra on a Banach space then there exists an isomorphism ψ from A onto B such that ?=kψ. For a nest algebra A in a factor von Neumann algebra, we characterize the linear maps on A such that δ(x)y+xδ(y)=0 for all x,y∈A with xy=0.     

Graphs with small boundary

For a pair of vertices x and y in a graph G, we denote by d_G(x,y) the distance between x and y in G. We call x a boundary vertex of y if x and y belong to the same component and d_G(y,v)?d_G(y,x) for each neighbor v of x in G. A boundary vertex of some vertex is simply called a boundary vertex, and the set of boundary vertices in G is called the boundary of G, and is denoted by B(G).In this paper, we investigate graphs with a small boundary. Since a pair of farthest vertices are boundary vertices, |B(G)|?2 for every connected graph G of order at least two. We characterize the graphs with boundary of order at most three. We cannot give a characterization of graphs with exactly four boundary vertices, but we prove that such graphs have minimum degree at most six. Finally, we give an upper bound to the minimum degree of a connected graph G in terms of |B(G)|.     

Embedding paratopological groups into topological products

We show that a Hausdorff paratopological group G admits a topological embedding as a subgroup into a topological product of Hausdorff first-countable (second-countable) paratopological groups if and only if G is ω-balanced (totally ω-narrow) and the Hausdorff number of G is countable, i.e., for every neighbourhood U of the neutral element e of G there exists a countable family γ of neighbourhoods of e such that _?V∈γVV−1⊆U. Similarly, we prove that a regular paratopological group G can be topologically embedded as a subgroup into a topological product of regular first-countable (second-countable) paratopological groups if and only if G is ω-balanced (totally ω-narrow) and the index of regularity of G is countable.As a by-product, we show that a regular totally ω-narrow paratopological group with countable index of regularity is Tychonoff.