# Signed Domination Number of the Directed Cylinder

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Signed Domination Number of ${P}_{m}\times {C}_{n}$

**Lemma**

**1.**

- (1)
- ${X}_{1}\subseteq P$.
- (2)
- For$2\le j\le m$,$|{X}_{j}\cap M|\le \lfloor n/2\rfloor $.
- (3)
- For$2\le j\le m-1$,$|({X}_{j}\cup {X}_{j+1})\cap M|\le n-\lceil n/3\rceil $.
- (4)
- For$2\le j\le m-2$,$\left|\right({X}_{j}\cup {X}_{j+1}\cup {X}_{j+2})\cap M|\le n$.

**Proof.**

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Case 1.**$m\equiv 0\phantom{\rule{3.33333pt}{0ex}}\left(mod\phantom{\rule{3.33333pt}{0ex}}3\right)$. Then $m=3r$ for some integer $r\ge 1$. We first show that ${\gamma}_{s}({P}_{m}\times {C}_{n})\le n(m+2)/3$ by constructing a SDF of ${P}_{m}\times {C}_{n}$ with weight $n(m+2)/3$. Let ${X}_{3j-1}^{*}={\cup}_{t=0}^{k-1}\left\{({u}_{3j-1},{v}_{3t+1})\right\}$, ${X}_{3j}^{*}={\cup}_{t=0}^{k-1}\left\{({u}_{3j},{v}_{3t+2})\right\}$ for $j=1,2,\dots ,r$, and ${X}_{3j+1}^{*}={\cup}_{t=0}^{k-1}\left\{({u}_{3j+1},{v}_{3t+3})\right\}$ for $j=1,2,\dots ,r-1$. Define $f:V({P}_{3r}\times {C}_{3k})\to \{-1,1\}$ by assigning to each vertex of $\left({\cup}_{j=1}^{r-1}\{{X}_{3j-1}^{*},{X}_{3j}^{*},{X}_{3j+1}^{*}\}\right)\cup \{{X}_{3r-1}^{*},{X}_{3r}^{*}\}$ the value $-1$ while to each vertex of $V({P}_{3r}\times {C}_{3k})\setminus \left({\cup}_{j=1}^{r-1}\{{X}_{3j-1}^{*},{X}_{3j}^{*},{X}_{3j+1}^{*}\}\right)\cup \{{X}_{3r-1}^{*},{X}_{3r}^{*}\}$ the value 1. It is easy to verify that f is a SDF of ${P}_{3r}\times {C}_{3k}$ with weight $w\left(f\right)=k(3r+2)=n(m+2)/3$ (The SDF f of ${P}_{9}\times {C}_{9}$ is shown in Figure 3). Thus ${\gamma}_{s}({P}_{3r}\times {C}_{3k})\le w\left(f\right)=n(m+2)/3$. By Lemma 1 (3) and (4), we have $\left|\right({X}_{2}\cup {X}_{3})\cap M|\le 2k$ and $\left|\right({X}_{3j-2}\cup {X}_{3j-1}\cup {X}_{3j})\cap M|\le 3k$ for every $2\le j\le r$, and so $\left|M\right|\le 2k+3k(r-1)=k(3r-1)$ from Lemma 1 (1). Then ${\gamma}_{s}({P}_{3r}\times {C}_{3k})=9rk-2\left|M\right|\ge 3rk+2k=n(m+2)/3$, which implies that ${\gamma}_{s}({P}_{3r}\times {C}_{3k})=n(m+2)/3$.

**Case 2.**$m\equiv 1\phantom{\rule{3.33333pt}{0ex}}\left(mod\phantom{\rule{3.33333pt}{0ex}}3\right)$. Then $m=3r+1$ for some integer $r\ge 1$. We first prove that ${\gamma}_{s}({P}_{m}\times {C}_{n})\le n(m+2)/3$ by establishing a SDF of ${P}_{m}\times {C}_{n}$ with weight $n(m+2)/3$. Set ${X}_{3j-1}^{*}={\cup}_{t=0}^{k-1}\left\{({u}_{3j-1},{v}_{3t+1})\right\}$, ${X}_{3j}^{*}={\cup}_{t=0}^{k-1}\left\{({u}_{3j},{v}_{3t+2})\right\}$ and ${X}_{3j+1}^{*}={\cup}_{t=0}^{k-1}\left\{({u}_{3j+1},{v}_{3t+3})\right\}$ for $j=1,2,\dots ,r$. Define $g:V({P}_{3r+1}\times {C}_{3k})\to \{-1,1\}$ as follows: each vertex of ${\cup}_{j=1}^{r}\{{X}_{3j-1}^{*},{X}_{3j}^{*},{X}_{3j+1}^{*}\}$ is assigned the value $-1$ while each vertex of $V({P}_{3r+1}\times {C}_{3k})\setminus {\cup}_{j=1}^{r}\{{X}_{3j-1}^{*},{X}_{3j}^{*},{X}_{3j+1}^{*}\}$ is assigned the value 1. It is not hard to see that g is a SDF of ${P}_{3r+1}\times {C}_{3k}$ with weight $w\left(g\right)=3k(r+1)=n(m+2)/3$ (The SDF g of ${P}_{7}\times {C}_{9}$ can be obtained by restricting f defined in Case 1 on ${P}_{7}\times {C}_{9}$, which is depicted in Figure 3). Thus ${\gamma}_{s}({P}_{3r+1}\times {C}_{3k})\le w\left(g\right)=n(m+2)/3$. For each $1\le j\le r$, it follows from Lemma 1 (4) that $\left|\right({X}_{3j-1}\cup {X}_{3j}\cup {X}_{3j+1})\cap M|\le 3k$. According to Lemma 1 (1), we have $\left|M\right|\le 3rk$. Hence ${\gamma}_{s}({P}_{3r+1}\times {C}_{3k})=3k(3r+1)-2\left|M\right|\ge 3rk+3k=n(m+2)/3$. This implies that ${\gamma}_{s}({P}_{3r+1}\times {C}_{3k})=n(m+2)/3$.

**Case 3.**$m\equiv 2\phantom{\rule{3.33333pt}{0ex}}\left(mod\phantom{\rule{3.33333pt}{0ex}}3\right)$. Then $m=3r+2$ for some integer $r\ge 1$. We first demonstrate that ${\gamma}_{s}({P}_{m}\times {C}_{n})\le n(m+2)/3$ by giving a SDF of ${P}_{m}\times {C}_{n}$ with weight $n(m+2)/3$. We write ${X}_{3j-1}^{*}={\cup}_{t=0}^{k-1}\left\{({u}_{3j-1},{v}_{3t+1})\right\}$ for $j=1,2,\dots ,r+1$, ${X}_{3j}^{*}={\cup}_{t=0}^{k-1}\left\{({u}_{3j},{v}_{3t+2})\right\}$ and ${X}_{3j+1}^{*}={\cup}_{t=0}^{k-1}\left\{({u}_{3j+1},{v}_{3t+3})\right\}$ for $j=1,2,\dots ,r$. Assigning to all vertices of $\left({\cup}_{j=1}^{r}\{{X}_{3j-1}^{*},{X}_{3j}^{*},{X}_{3j+1}^{*}\}\right)\cup \left\{{X}_{3r+2}^{*}\right\}$ the value $-1$ and to all other vertices the value 1, we produce a SDF h of ${P}_{3r+2}\times {C}_{3k}$ with weight $w\left(f\right)=k(3r+4)=n(m+2)/3$ (The SDF h of ${P}_{8}\times {C}_{9}$ can be obtained by restricting f defined in Case 1 (see Figure 3) to ${P}_{8}\times {C}_{9}$). So ${\gamma}_{s}({P}_{3r+2}\times {C}_{3k})\le n(m+2)/3$. According to Lemma 1 (3) and (4), we obtain $\left|\right({X}_{2}\cup {X}_{3})\cap M|\le 2k$, $\left|\right({X}_{3r+1}\cup {X}_{3r+2})\cap M|\le 2k$ and $\left|\right({X}_{3j-2}\cup {X}_{3j-1}\cup {X}_{3j})\cap M|\le 3k$ for every $2\le j\le r$. This means that $\left|M\right|\le 4k+3k(r-1)=k(3r+1)$ by Lemma 1 (1). Therefore, ${\gamma}_{s}({P}_{3r+2}\times {C}_{3k})=3k(3r+2)-2\left|M\right|\ge 3rk+4k=n(m+2)/3$, implying that ${\gamma}_{s}({P}_{3r+2}\times {C}_{3k})=n(m+2)/3$. This completes the proof of Theorem 2. □

**Theorem**

**3.**

**Proof.**

**Case 1.**$m\equiv 0\phantom{\rule{3.33333pt}{0ex}}\left(mod\phantom{\rule{3.33333pt}{0ex}}2\right)$. Then $m=2r$ for some integer $r\ge 2$. We first show that ${\gamma}_{s}({P}_{m}\times {C}_{n})\le (m(n+2)+2n-8)/3$ by constructing a SDF of ${P}_{m}\times {C}_{n}$ with weight $\left(m\right(n+2)+2n-8)/3$. Set ${X}_{2j-1}^{*}={\cup}_{t=0}^{k-2}\left\{({u}_{2j-1},{v}_{n+2-j-3t})\right\}$ for $j=2,3,\dots ,r$ and ${X}_{2j}^{*}={\cup}_{t=0}^{k-1}\left\{({u}_{2j},{v}_{n+3-j-3t})\right\}\cup \left\{({u}_{2j},{v}_{5-j})\right\}$ for $j=1,2,\dots ,r$. Define $g:V({P}_{2r}\times {C}_{3k+1})\to \{-1,1\}$ as follow: each vertex of ${\cup}_{j=2}^{r}\{{X}_{2j-1}^{*},{X}_{2j}^{*}\}\cup ({\cup}_{t=0}^{k-1}\left\{({u}_{2},{v}_{n+2-3t})\right\}\cup \left\{({u}_{2},{v}_{4})\right\})$ is assigned the value $-1$ while each vertex of $V({P}_{2r}\times {C}_{3k+1})\setminus ({\cup}_{j=2}^{r}\{{X}_{2j-1}^{*},{X}_{2j}^{*}\}\cup ({\cup}_{t=0}^{k-1}\left\{({u}_{2},{v}_{n+2-3t})\right\}\cup \left\{({u}_{2},{v}_{4})\right\}))$ is assigned the value 1. It is not hard to verify that g is a SDF of ${P}_{2r}\times {C}_{3k+1}$ with weight $w\left(g\right)=2r(k+1)+2k-2=\left(m\right(n+2)+2n-8)/3$ (The SDF g of ${P}_{8}\times {C}_{10}$ is shown in Figure 4). Thus ${\gamma}_{s}({P}_{2r}\times {C}_{3k+1})\le w\left(g\right)=(m(n+2)+2n-8)/3$. By Lemma 1 (3) and (4), we have $\left|\right({X}_{2}\cup {X}_{3}\cup {X}_{4})\cap M|\le 3k+1$ and $\left|\right({X}_{2j-1}\cup {X}_{2j})\cap M|\le 2k$ for every $3\le j\le r$, and so $\left|M\right|\le (3k+1)+2k(r-2)=k(2r-1)+1$ from Lemma 1 (1). Then ${\gamma}_{s}({P}_{2r}\times {C}_{3k+1})=2r(3k+1)-2\left|M\right|\ge 2r(k+1)+2k-2=(m(n+2)+2n-8)/3$, which implies that ${\gamma}_{s}({P}_{2r}\times {C}_{3k+1})=(m(n+2)+2n-8)/3$.

**Case 2.**$m\equiv 1\phantom{\rule{3.33333pt}{0ex}}\left(mod\phantom{\rule{3.33333pt}{0ex}}2\right)$. Then $m=2r+1$ for some integer $r\ge 1$. We first demonstrate that ${\gamma}_{s}({P}_{m}\times {C}_{n})\le (m(n+2)+2n-2)/3$ by establishing a SDF of ${P}_{m}\times {C}_{n}$ with weight $\left(m\right(n+2)+2n-2)/3$. We write ${X}_{2j-1}^{*}={\cup}_{t=0}^{k-2}\left\{({u}_{2j-1},{v}_{n+2-j-3t})\right\}$ for $j=2,3,\dots ,r+1$ and ${X}_{2j}^{*}={\cup}_{t=0}^{k-1}\left\{({u}_{2j},{v}_{n+3-j-3t})\right\}\cup \left\{({u}_{2j},{v}_{5-j})\right\}$ for $j=1,2,\dots ,r$. Assigning to all vertices of ${\cup}_{j=2}^{r}\{{X}_{2j-1}^{*},{X}_{2j}^{*}\}\cup ({\cup}_{t=0}^{k-1}\left\{({u}_{2},{v}_{n+2-3t})\right\}\cup \left\{({u}_{2},{v}_{4})\right\})\cup \left({\cup}_{t=0}^{k-2}\left\{({u}_{2r+1},{v}_{n+1-r-3t})\right\}\right)$ the value $-1$, and to all other vertices the value 1, we produce a SDF h of ${P}_{2r+1}\times {C}_{3k+1}$ with weight $w\left(h\right)=2r(k+1)+3k+1=\left(m\right(n+2)+2n-2)/3$ (The SDF h of ${P}_{7}\times {C}_{10}$ can be obtained by restricting g defined in Case 1 (see Figure 4) to ${P}_{7}\times {C}_{10}$). So ${\gamma}_{s}({P}_{2r+1}\times {C}_{3k+1})\le w\left(h\right)=(m(n+2)+2n-2)/3$. For every $1\le j\le r$, it follows from Lemma 1 (3) that $\left|\right({X}_{2j}\cup {X}_{2j+1})\cap M|\le 2k$. This derives that $\left|M\right|\le 2kr$ by Lemma 1 (1). Therefore, ${\gamma}_{s}({P}_{2r+1}\times {C}_{3k+1})=(2r+1)(3k+1)-2\left|M\right|\ge 2r(k+1)+3k+1=(m(n+2)+2n-2)/3$, implying that ${\gamma}_{s}({P}_{2r+1}\times {C}_{3k+1})=(m(n+2)+2n-2)/3$. This completes the proof of Theorem 3. □

**Theorem**

**4.**

**Proof.**

**Case 1.**$m\equiv 0\phantom{\rule{3.33333pt}{0ex}}\left(mod\phantom{\rule{3.33333pt}{0ex}}2\right)$. Then $m=2r$ for some integer $r\ge 2$. We first prove that ${\gamma}_{s}({P}_{m}\times {C}_{n})\le (m(n+1)+2n-4)/3$ by constructing a SDF of ${P}_{m}\times {C}_{n}$ with weight $\left(m\right(n+1)+2n-4)/3$. Let ${X}_{2j-1}^{*}={\cup}_{t=0}^{k-1}\left\{({u}_{2j-1},{v}_{n+2-j-3t})\right\}$ for $j=2,3,\dots ,r$ and ${X}_{2j}^{*}={\cup}_{t=0}^{k}\left\{({u}_{2j},{v}_{n+3-j-3t})\right\}$ for $j=1,2,\dots ,r$. Define $g:V({P}_{2r}\times {C}_{3k+2})\to \{-1,1\}$ as follow: each vertex of ${\cup}_{j=2}^{r}\{{X}_{2j-1}^{*},{X}_{2j}^{*}\}\cup \left({\cup}_{t=0}^{k}\left\{({u}_{2},{v}_{n+2-3t})\right\}\right)$ is assigned the value $-1$ while each vertex of $V({P}_{2r}\times {C}_{3k+2})\setminus {\cup}_{j=2}^{r}\{{X}_{2j-1}^{*},{X}_{2j}^{*}\}\cup \left({\cup}_{t=0}^{k}\left\{({u}_{2},{v}_{n+2-3t})\right\}\right)$ is assigned the value 1. It is not hard to verify that g is a SDF of ${P}_{2r}\times {C}_{3k+2}$ with weight $w\left(g\right)=2r(k+1)+2k=\left(m\right(n+1)+2n-4)/3$ (The SDF g of ${P}_{8}\times {C}_{11}$ is shown in Figure 5). Thus ${\gamma}_{s}({P}_{2r}\times {C}_{3k+2})\le w\left(g\right)=(m(n+1)+2n-4)/3$. By Lemma 1 (3) and (4), we have $\left|\right({X}_{2}\cup {X}_{3}\cup {X}_{4})\cap M|\le 3k+2$ and $\left|\right({X}_{2j-1}\cup {X}_{2j})\cap M|\le 2k+1$ for every $3\le j\le r$. So $\left|M\right|\le (3k+2)+(2k+1)(r-2)=k(2r-1)+r$ according to Lemma 1 (1). Then ${\gamma}_{s}({P}_{2r}\times {C}_{3k+2})=2r(3k+2)-2\left|M\right|\ge 2r(k+1)+2k=(m(n+1)+2n-4)/3$, which implies that ${\gamma}_{s}({P}_{2r}\times {C}_{3k+2})=(m(n+1)+2n-4)/3$.

**Case 2.**$m\equiv 1\phantom{\rule{3.33333pt}{0ex}}\left(mod\phantom{\rule{3.33333pt}{0ex}}2\right)$. Then $m=2r+1$ for some integer $r\ge 1$. We first demonstrate that ${\gamma}_{s}({P}_{m}\times {C}_{n})\le (m(n+1)+2n-1)/3$ by establishing a SDF of ${P}_{m}\times {C}_{n}$ with weight $\left(m\right(n+1)+2n-1)/3$. We write ${X}_{2j-1}^{*}={\cup}_{t=0}^{k-1}\left\{({u}_{2j-1},{v}_{n+2-j-3t})\right\}$ for $j=2,3,\dots ,r+1$ and ${X}_{2j}^{*}={\cup}_{t=0}^{k}\left\{({u}_{2j},{v}_{n+3-j-3t})\right\}$ for $j=1,2,\dots ,r$. Assigning to all vertices of ${\cup}_{j=2}^{r}\{{X}_{2j-1}^{*},{X}_{2j}^{*}\}\cup \left({\cup}_{t=0}^{k-1}\left\{({u}_{2},{v}_{n+2-3t})\right\}\right)\cup \left({\cup}_{t=0}^{k-1}\left\{({u}_{2r+1},{v}_{n+1-r-3t})\right\}\right)$ the value $-1$, and to all other vertices the value 1, we produce a SDF h of ${P}_{2r+1}\times {C}_{3k+2}$ with weight $w\left(h\right)=2r(k+1)+3k+2=\left(m\right(n+1)+2n-1)/3$ (The SDF h of ${P}_{7}\times {C}_{11}$ can be obtained by restricting g defined in Case 1 (see Figure 5) to ${P}_{7}\times {C}_{11}$). So ${\gamma}_{s}({P}_{2r+1}\times {C}_{3k+2})\le w\left(h\right)=(m(n+1)+2n-1)/3$. For every $1\le j\le r$, it follows from Lemma 1 (3) that $\left|\right({X}_{2j}\cup {X}_{2j+1})\cap M|\le 2k+1$. This means that $\left|M\right|\le r(2k+1)$ by Lemma 1 (1). Therefore, ${\gamma}_{s}({P}_{2r+1}\times {C}_{3k+2})=(2r+1)(3k+2)-2\left|M\right|\ge 2r(k+1)+3k+2=(m(n+1)+2n-1)/3$, implying that ${\gamma}_{s}({P}_{2r+1}\times {C}_{3k+2})=(m(n+1)+2n-1)/3$. This completes the proof of Theorem 4. □

## 3. Conclusions and Future Work

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Chartrand, G.; Lesniak, L. Graphs and Digraphs, 4th ed.; Chapman and Hall: Boca Raton, FL, USA, 2005. [Google Scholar]
- Martínez, A.C.; Montejano, L.P.; Rodríguez-Velázquez, J.A. Total weak Roman domination in graphs. Symmetry
**2019**, 11, 831. [Google Scholar] [CrossRef] - Martínez, A.C.; Montejano, L.P.; Rodríguez-Velázquez, J.A. On the secure total domination number of graphs. Symmetry
**2019**, 11, 1165. [Google Scholar] [CrossRef] - Haynes, T.W.; Hedetniemi, S.T.; Slater, P.J. Domination in Graphs: Advanced Topics; Marcel Dekker: New York, NY, USA, 1998. [Google Scholar]
- Haynes, T.W.; Hedetniemi, S.T.; Slater, P.J. Foundamentals of Domination in Graphs; Marcel Dekker: New York, NY, USA, 1998. [Google Scholar]
- Dunbar, J.E.; Hedetniemi, S.T.; Henning, M.A.; Slater, P.J. Signed Domination in Graphs, Graph Theory, Combinatorics, and Applications; John Wiley Sons, Inc.: Hoboken, NJ, USA, 1995; Volume 1, pp. 311–322. [Google Scholar]
- Chen, W.; Song, E. Lower bounds on several versions of signed domination number. Discrete Math.
**2008**, 308, 1837–1846. [Google Scholar] [CrossRef] - Favaron, O. Signed domination in regular graphs. Discrete Math.
**1996**, 158, 287–293. [Google Scholar] [CrossRef] - Füredi, Z.; Mubayi, D. Signed domination in regular graphs and set-systems. J. Combin. Theory Ser. B
**1999**, 76, 223–239. [Google Scholar] [CrossRef] - Henning, M.A. Signed total domination in graphs. Discrete Math.
**2004**, 278, 109–125. [Google Scholar] [CrossRef] - Matoušek, J. On the signed domination in graphs. Combinatorica
**2000**, 20, 103–108. [Google Scholar] [CrossRef] - Mishra, S. Complexity of majority monopoly and signed domination problems. J. Discrete Algorithms
**2012**, 10, 49–60. [Google Scholar] [CrossRef] - Shan, E.F.; Cheng, T.C.E. Remarks on the minus (signed) total domination in graphs. Discrete Math.
**2008**, 308, 3373–3380. [Google Scholar] [CrossRef] - Shan, E.F.; Cheng, T.C.E.; Kang, L.Y. An application of the Turán theorem to domination in graphs. Discrete Appl. Math.
**2008**, 156, 2712–2718. [Google Scholar] [CrossRef] - Sohn, M.Y.; Lee, J.; Kwon, Y.S. Lower bounds of signed domination number of a graph. Bull. Korean Math. Soc.
**2004**, 41, 181–188. [Google Scholar] [CrossRef] - Zelinka, B. Signed domination numbers of directed graphs. Czechoslov. Math. J.
**2005**, 55, 479–482. [Google Scholar] [CrossRef] - Karami, H.; Sheikholeslami, S.M.; Khodkar, A. Lower bounds on the signed domination numbers of directed graphs. Discrete Math.
**2009**, 309, 2567–2570. [Google Scholar] [CrossRef] - Volkmann, L. Signed domination and signed domatic numbers of digraphs. Discuss. Math. Graph Theory
**2011**, 31, 415–427. [Google Scholar] [CrossRef] - Hammack, R.; Imrich, W.; Klavžar, S. Handbook of Product Graphs, 2nd ed.; CRC Press: Boca Raton, FL, USA, 2011. [Google Scholar]
- Kumar, N.; Kumar, M.; Patel, R.B. Capacity and interference aware link scheduling with channel assignment in wireless mesh networks. J. Netw. Comput. Appl.
**2011**, 34, 30–38. [Google Scholar] [CrossRef] - Crevals, S.; Wang, H.C.; Kim, H.K.; Baek, H. Domination number of the directed cylinder. Australas. J. Comb.
**2015**, 61, 192–209. [Google Scholar] - Liu, J.; Zhang, X.D.; Chen, X.; Meng, J.X. The domination number of Cartesian products of directed cycles. Inform. Process. Lett.
**2010**, 110, 171–173. [Google Scholar] [CrossRef] - Liu, J.; Zhang, X.D.; Meng, J.X. On domination number of Cartesian product of directed paths. J. Combin. Optim.
**2011**, 22, 651–662. [Google Scholar] [CrossRef] - Mollard, M. On domination of Cartesian product of directed cycles. Discuss. Math. Graph Theory
**2013**, 33, 387–394. [Google Scholar] [CrossRef] - Mollard, M. The domination number of Cartesian product of two directed paths. J. Combin. Optim.
**2014**, 27, 144–151. [Google Scholar] [CrossRef] - Shaheen, R. Domination number of toroidal grid digraphs. Util. Math.
**2009**, 78, 175–184. [Google Scholar] - Zhang, X.D.; Liu, J.; Chen, X.; Meng, J.X. Domination number of Cartesian products of directed cycles. Inform. Process. Lett.
**2010**, 111, 36–39. [Google Scholar] [CrossRef] - Zhang, Z.; Shaheen, R. On signed domination number of Cartesian product of directed paths. Asian J. Math. Comput. Res.
**2017**, 18, 113–119. [Google Scholar] - Wang, H.C.; Kim, H.K.; Deng, Y.P. On signed domination number of Cartesian product of directed paths. Util. Math.
**2018**, 109, 45–61. [Google Scholar]

**Figure 3.**The SDF f of ${P}_{9}\times {C}_{9}$. The red solid vertices are assigned the value $-1$ under f.

**Figure 4.**The SDF g of ${P}_{8}\times {C}_{10}$. The red solid vertices are assigned the value $-1$ under g.

**Figure 5.**The SDF g of ${P}_{8}\times {C}_{11}$. The red solid vertices are assigned the value $-1$ under g.

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Wang, H.; Kim, H.K.
Signed Domination Number of the Directed Cylinder. *Symmetry* **2019**, *11*, 1443.
https://doi.org/10.3390/sym11121443

**AMA Style**

Wang H, Kim HK.
Signed Domination Number of the Directed Cylinder. *Symmetry*. 2019; 11(12):1443.
https://doi.org/10.3390/sym11121443

**Chicago/Turabian Style**

Wang, Haichao, and Hye Kyung Kim.
2019. "Signed Domination Number of the Directed Cylinder" *Symmetry* 11, no. 12: 1443.
https://doi.org/10.3390/sym11121443