The maximum forcing number of polyomino

The forcing number of a perfect matching $M$ of a graph $G$ is the cardinality of the smallest subset of $M$ that is contained in no other perfect matchings of $G$. For a planar embedding of a 2-connected bipartite planar graph $G$ which has a perfect matching, the concept of Clar number of hexagonal system had been extended by Abeledo and Atkinson as follows: a spanning subgraph $C$ of is called a Clar cover of $G$ if each of its components is either an even face or an edge, the maximum number of even faces in Clar covers of $G$ is called Clar number of $G$, and the Clar cover with the maximum number of even faces is called the maximum Clar cover. It was proved that if $G$ is a hexagonal system with a perfect matching $M$ and $K'$ is a set of hexagons in a maximum Clar cover of $G$, then $G-K'$ has a unique 1-factor. Using this result, Xu {\it et. at.} proved that the maximum forcing number of the elementary hexagonal system are equal to their Clar numbers, and then the maximum forcing number of the elementary hexagonal system can be computed in polynomial time. In this paper, we show that an elementary polyomino has a unique perfect matching when removing the set of tetragons from its maximum Clar cover. Thus the maximum forcing number of elementary polyomino equals to its Clar number and can be computed in polynomial time. Also, we have extended our result to the non-elementary polyomino and hexagonal system

Resistive Pulse study of Vesicles and Liposomes

In this work, the properties of the liposomes, the artificially created vesicles by various methods, are explored by a resistive pulse method using micropipettes. The fact that vesicles are fundamental in the wide range of functionalities they fulfill as organelles strengthen the desire of understanding the properties of them. The motivation of this work comes from the significant roles that liposomes play in the development of targeted drug delivery systems. Among other significant variables, the size of liposomes is found to be one of the dominating parameters in liposome based drug delivery, and the correlation between liposome size and delivery efficiency is discussed. To help improving the size evaluation ability, a few mainstream methods for liposome size detection and measurements are reviewed. As a reliable and accessible alternative method for liposomes detection, the resistive pulse method is introduced and the measurement on liposomes size change upon pH gradient was performed using this method. With our current liposome composition, we found the size increases as environmental pH increases. Further investigation is performed with vesicular pH=6, 7, and 8, respectively. Lastly, the stability of the small unilamellar vesicles (SUV) was studied via resistive pulse method, by monitoring the size change of 50nm liposomes as function of time. A significant size change in freshly prepared 50nm liposomes is recorded. This information will provide invaluable knowledge for targeting tumor with tight tissues, where small size liposomes are needed