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 a connected graph of order $n$, whose minimum vertex degree is atleast $k$. A subset $S$ of vertices in $G$ is a $k$-tuple total dominating setif every vertex of $G$ is adjacent to at least $k$ vertices in $S$. The minimumcardinality of a $k$-tuple total dominating set of $G$ is the $k$-tuple totaldomination number of $G$, denoted by $\gamma_{\times k,t}(G)$. Henning and Yeoin \cite{hen} proved that if $G$ is a cubic graph different from the Heawoodgraph, $\gamma_{\times 2, t}(G) \leq \frac{5}{6}n$, and this bound is sharp.Similarly, a $k$-tuple dominating set is a subset $S$ of vertices of $G$, $V(G)$ such that $|N[v] \cap S| \geq k$ for every vertex $v$, where $N[v] =\{v\}\cup \{u \in V(G) : uv \in E(G)\}$. The $k$-tuple domination number of$G$, denoted by $\gamma_{\times k}(G)$, is the minimum cardinality of a$k$-tuple dominating set of $G$.  In this paper, we give a simple approach to compute an upper bound for$(r-1)$-tuple total domination number of $r$-regular graphs. Also, we give anupper bound for the $r$-tuple dominating number of $r$-regular graphs. Inaddition, our method gives algorithms to compute dominating sets with the givenbounds, while the previous methods are existential.

On $k$-tuple and $k$-tuple total domination numbers of regular graphs