English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

Consider a graph each of whose vertices is either in the ON state or in the OFF state and call  the resulting ordered bipartition into ON vertices and OFF vertices a configuration of the graph.   A regular move at a vertex changes the states of the neighbors of that vertex and hence sends the current configuration to another one. A valid move is a regular move at an ON vertex.   For any graph $G,$ let $\mathcal{D}(G)$   be the minimum integer such that given any starting configuration $\bf x$ of $G$ there must exist a sequence of valid moves which takes $\bf x$    to a configuration with at most $\ell +\mathcal{D}(G)$ ON verticesprovided there is a sequence of regular moves which brings $\bf x$ to a  configuration in which there are $\ell$ ON vertices.  The shadow graph $\mathcal{S}(G)$ of a graph $G$ is obtained from $G$ by deleting all loops. We prove that  $\mathcal{D}(G)\leq 3$ if $\mathcal{S}(G)$ is unicyclic and give an example to show that the bound $3$ is tight. We also prove that $\mathcal{D}(G)\leq 2$ if $ G $ is a two-dimensional grid graph and  $\mathcal{D}(G)=0$ if $\mathcal{S}(G)$ is a two-dimensional grid graph but not a path  and $G\neq  \mathcal{S}(G)$.

Minimum Light Numbers in the $\sigma$-Game and Lit-Only $\sigma$-Game on Unicyclic and Grid Graphs