The 2-Disjoint Connected Subgraphs problem asks if a given graph has two vertex-disjoint connected subgraphs containing prespecified sets of vertices. We show that this problem is NP-complete even if one of the sets has cardinality 2. The Longest Path Contractibility problem asks for the largest integer ` for which an input graph can be contracted to the path P` on ` vertices. We show that the ...

In this paper, we nd the relationships between module contractibility of aBanach algebra and its ideals. We also prove that module contractibility ofa Banach algebra is equivalent to module contractibility of its module uniti-zation. Finally, we show that when a maximal group homomorphic image ofan inverse semigroup S with the set of idempotents E is nite, the moduleprojective tensor product l1...

In this note, we show that cite[Corollary 3.2]{sad} is not always true. In fact, we characterize essential left $phi$-contractibility of the group algebras in terms of compactness of its related locally compact group. Also, we show that for any compact commutative group $G$, $L^{2}(G)$ is always essentially left $phi$-contractible. We discuss the essential left $phi$-contractibility of some Fou...

let s be a locally compact foundation semigroup with identity and                          be its semigroup algebra. let x be a weak*-closed left translation invariant subspace of    in this paper, we prove that  x  is invariantly  complemented in   if and  only if  the left ideal  of    has a bounded approximate identity. we also prove that a foundation semigroup with identity s is left amenab...

In this paper we study the concept of ph-biatness ofa Banach algebra A, where ph is a continuous homomorphism on A.We prove that if ph is a continuous epimorphism on A and A hasa bounded approximate identity and A is ph- biat, then A is ph-amenable. In the case where ph is an isomorphism on A we showthat the ph- amenability of A implies its ph-biatness.

we survey the recent investigations on (bounded, sequential) approximate connesamenability and pseudo-connes amenability for dual banach algebras. we will discuss thecore problems concerning these notions and address the signicance of any solutions to themto the development of the eld.

We study the global dimension function $${\text {gldim}}:{\text {Aut}}\backslash {\text {Stab}}{\mathcal {D}}/\mathbb {C}\rightarrow \mathbb {R}_{\ge 0}$$ on quotient of space Bridgeland stability conditions a triangulated category $${\mathcal {D}}$$ as well Toda’s Gepner equation $$\Phi (\sigma )=s\cdot \sigma $$ for some $$\sigma \in and $$(\Phi ,s)\in {Aut}}{\mathcal {D}}\times {C}$$ . For b...

IT-enabled exchange in electronic markets has significant implications for buyer-supplier relationships. This paper builds on transaction cost and routine based perspectives in analyzing buyers’ decisions to use reverse auctions, which are an emerging IT-enabled exchange mechanism. The study argues that buyers are less likely to adopt reverse auctions for supplier relationships with high degree...