The Price of Connectivity for Vertex Cover

The vertex cover number of a graph is the minimum number of vertices that are needed to cover all edges. When those vertices are further required to induce a connected subgraph, the corresponding number is called the connected vertex cover number, and is always greater or equal to the vertex cover number. Connected vertex covers are found in many applications, and the relationship between those two graph invariants is therefore a natural question to investigate. For that purpose, we introduce the {\em Price of Connectivity}, defined as the ratio between the two vertex cover numbers. We prove that the price of connectivity is at most 2 for arbitrary graphs. We further consider graph classes in which the price of connectivity of every induced subgraph is bounded by some real number $t$. We obtain forbidden induced subgraph characterizations for every real value $t \leq 3/2$. We also investigate critical graphs for this property, namely, graphs whose price of connectivity is strictly greater than that of any proper induced subgraph. Those are the only graphs that can appear in a forbidden subgraph characterization for the hereditary property of having a price of connectivity at most $t$. In particular, we completely characterize the critical graphs that are also chordal. Finally, we also consider the question of computing the price of connectivity of a given graph. Unsurprisingly, the decision version of this question is NP-hard. In fact, we show that it is even complete for the class $\Theta_2^P = P^{NP[\log]}$, the class of decision problems that can be solved in polynomial time, provided we can make $O(\log n)$ queries to an NP-oracle. This paves the way for a thorough investigation of the complexity of problems involving ratios of graph invariants.Comment: 19 pages, 8 figure

SPLITTING AND AUTHORISING CARD PAYMENTS ACROSS MULTIPLE USERS ONLINE VIA A SINGLE PAYMENT TOKEN

The present disclosure relates to a method for splitting and authorizing payments online using tokens. The present disclosure demonstrates how a token based on card or payment account reference of a cardholder can be used to trigger payments from multiple accounts to split the total transaction cost based on a set of pre-determined rules. The method allows users to link multiple payment accounts to a single lead token, which in turn can be used to make payments from said accounts according to a set of rules. Generating a single lead token, allows this payment splitting method to be used at an existing card-based point of sale without the need for multi-party ecosystem enablement. The method can also enable dual/ multiple authorizations of a payment based on the lead token, for example a payment card, by requesting further authorization from another party. The method works by mapping the generated token, rules, and the payment account, for example a Primary Account Number (PAN) together in a lookup fashion to be read and processed by an authorization system. Based on the rules associated with the tokens, the authorization system initiates separate transactions and approves them or seeks further authorisation for a transaction. Initiating a single token payment, matching the format, and processing requirements of existing payment methods, triggering additional authorizations compliant with existing payment methods to receive appropriate authorizations from multiple accounts and then returning a single authorisation response tied to a single reference for the transaction, allows instant operability of the solution without the need for multi-party ecosystem enablement

Shifting Identities in the Internet Age

http://deepblue.lib.umich.edu/bitstream/2027.42/111656/1/samolive_1429898601.pd

PAYMENT CARD AS A UNIQUE IDENTIFIER FOR IDENTIFICATION AND DELAYED PAYMENT

The present invention provides a system and method for automatically identifying, assigning, aggregating and generating a bill for a customer using a unique identifier, where the unique identifier is a primary account number (PAN). The present system obtains the payment credentials through a payment device from the customer before placing the order or while ‘paying’ for the order and instead of immediately processing a standard payment, the system generates a hash value by performing a hash algorithm. The generated hash value is stored and or matched to the customer. The merchant or ePOS then assigns the order details to the corresponding hash value/ customer. When the customer wishes to pay for the bill, the payment device reads the payment credentials of the cardholder once again to generate a hash value and compares the generated hash value with the previously stored hash values. If the hash values are matched, the order details corresponding to that hash value are obtained and generates the single automatically reconciled bill payment. In an non limiting aspect, the system, upon receiving the PAN, may call an external server/database to receive additional data based on the PAN such as: the Personal Account reference (PAR) number or Cardholder name, etc

DRAWING DOWN ON THE VALUE OF A NON-FUNGIBLE TOKEN (NFT) AND ITS ASSOCIATED ASSET VIA SALE OR TRANSFER ON THE BLOCKCHAIN TO FUND AND OR PROVIDE CREDIT COLLATERAL ON A PURCHASE

The present disclosure relates to a method and system for drawing down on an NFT’s value to fund a purchase in real-time. The method comprises transmitting a purchase request to an NFT Application Programming Interface (API) 108 hosted on a payment server seeking an authorization to purchase goods/services using the value of the NFT. In response to the purchase request, the NFT API 108 holding the NFT may transfer, via sale on the blockchain the NFT’s ownership to the merchant to settle and approve the transaction. Alternatively, the NFT API 108 may sell a portion of the NFT to the market to convert the NFT to a fiat currency and process the transaction using the fiat currency. Alternatively, the NFT API 108 may sell a portion of the NFT to a financial institution which has bid the most competitive terms to provide a credit facility to the purchaser. The financial institution will hold this as collateral against the credit facility that is provided. Once the collateral has been received, the financial institution will provide the fiat currency required to process the transaction. Once the proportionate amount of the NFT has been sold to fund the purchase, ownership of the purchaser of the NFT is drawn down and the transaction is approved. Finally, the payment server sends an authorization to the merchant and transaction is completed

A SYSTEM AND METHOD FOR PROVIDING MUTIPLE TRASACTIONS FOR A SINGLE PAYMENT

Present disclosure provides a system and method for providing multiple transactions by tapping multiple cards consecutively on a Point of Sale (PoS) device (104) for a single payment. The system may include a PoS device (104) and a network server (109). The PoS device (104) receives merchant input for a payment. The merchant (102) input includes total transaction amount and payment split input into the PoS device (104). The payment split input indicates the total number of customers who desire to divide the total transaction amount within themselves. The PoS device (104) splits the total amount by the number input by the merchant (102). The PoS device (104) then receives payment card information related to each of the at least two customers (105, 106) based on the split calculation. After receiving all the payment card details, the PoS device (104) sends the card details to the network server (109) for authorisation. The network server (109) sends authorization requests related to each of the at least two customers (105, 106) to a respective acquirer entity via the communication network (108). Upon receiving authorization responses from the network server (109), the PoS device (104) generates a single receipt to the merchant including the multiple payment transactions details related to each of the at least two customers (105, 106). This reduces the idling time on the PoS device (104). Thereby, a quicker and efficient payment process may be achieved

Towards a unified analysis of the syntax and semantics of get constructions

PhDThis thesis develops a novel, unified, syntactic and semantic analysis for a range of get constructions including those with adjectival, prepositional and verbal complements. There are two reasons to believe that such an approach is justified. First, the relevant get constructions demonstrate similar semantic characteristics across complement types, e.g. the presence of Cause (in the sense of Pylkkanen 2008), leading to an obligatorily resultative change-of-state interpretation. Second, the range of constructions display syntactic similarities: for each get construction with no external argument there is a corresponding construction with an external argument; and all of the relevant get constructions take a predicative small clause complement. The approach defended here utilises a formal syntactic and semantic framework to propose an analysis in which get is interpreted as a causative functional head which takes a PredP complement whose function is to add a Holder argument to the property expression in its complement (Bowers 1993, Adger and Ramchand 2003). At this point one of two things may occur. Either (i), the Holder argument raises to the sentential subject position, or (ii), it remains in-situ and an argument external to the causative head is introduced, and then raised to subject position. The thesis shows that, contra Pylkk¨anen 2008, and unlike any other English constructions, get constructions may project Cause without necessarily ‘bundling’ it together with Voice in the syntax. The resulting claims impact on topics in theoretical linguistics as varied as predication, causation, reflexivity and binding, property theory and passivisation, and hold consequences for the nature of the syntax semantics interface