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

Let $G$ be any simple and undirected graph. By a threshold assignment $\tau$in $G$ we mean any function $\tau:V(G)\rightarrow \mathbb{N}$ such that$\tau(v)\leq d_G(v)$ for any vertex $v$ of $G$. Given a graph $G$ with athreshold assignment $\tau$, a subset of vertices $M$ is said to be a$\tau$-monopoly if there exist at least $\tau(v)$ neighbors in $M$ for anyvertex $v \in V(G) \setminus M$. Similarly, a subset of vertices $D$ is said tobe a $\tau$-dynamic monopoly if starting with the set $D$ and iterativelyadding to the current set further vertices $u$ that have at least $\tau(u)$neighbors in it, results in the entire vertex set $V(G)$. Denote by$mon_{\tau}(G)$ (resp. $dyn_{\tau}(G)$) the smallest cardinality of a$\tau$-monopoly (resp. $\tau$-dynamic monopoly) of the graph among all others.In this paper we obtain some lower and upper bounds for these two parameterswith constant threshold assignments for Cartesian product graphs. Our boundsimprove the previous known bounds. We also determine the exact value of thesetwo parameters with fixed thresholds in some Cartesian graph products includingcycles and complete graphs.

On monopoly and dynamic monopoly of Cartesian product of graphs with constant thresholds