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Total domination in cubic Knödel graphs | ||
| Communications in Combinatorics and Optimization | ||
| دوره 6، شماره 2، اسفند 2021، صفحه 221-230 اصل مقاله (417.76 K) | ||
| نوع مقاله: Original paper | ||
| شناسه دیجیتال (DOI): 10.22049/cco.2020.26793.1143 | ||
| نویسندگان | ||
| Nader Jafari Rad* 1؛ Doost Ali Mojdeh2؛ Reza Musawi3؛ E. Nazari4 | ||
| 1Shahed University | ||
| 2Departtment of Mathematics, University of Mazandaran | ||
| 3Shahrood University of Technology | ||
| 4Tafresh University | ||
| چکیده | ||
| A subset $D$ of vertices of a graph $G$ is a dominating set if for each $u\in V(G) \setminus D$, $u$ is adjacent to some vertex $v\in D$. The domination number, $\gamma(G)$ of $G$, is the minimum cardinality of a dominating set of $G$. A set $D\subseteq V(G)$ is a total dominating set if for each $u\in V(G)$, $u$ is adjacent to some vertex $v\in D$. The total domination number, $\gamma_{t}(G)$ of $G$, is the minimum cardinality of a total dominating set of $G$. For an even integer $n\ge 2$ and $1\le \Delta \le \lfloor\log_2n\rfloor$, a Kn\"odel graph $W_{\Delta,n}$ is a $\Delta$-regular bipartite graph of even order $n$, with vertices $(i,j)$, for $i=1,2$ and $0\le j\le \frac{n}{2}-1$, where for every $j$, $0\le j\le \frac{n}{2}-1$, there is an edge between vertex $(1, j)$ and every vertex $(2,(j+2^k-1)$ mod $\frac{n}{2}$), for $k=0,1,\dots,\Delta-1$. In this paper, we determine the total domination number in $3$-regular Kn\"odel graphs $W_{3,n}$. | ||
| کلیدواژهها | ||
| Knodel graph؛ domination number؛ total domination number؛ Pigeonhole Principle | ||
| مراجع | ||
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