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Volume 33, Issue 4
Signed Roman (Total) Domination Numbers of Complete Bipartite Graphs and Wheels

Yancai Zhao & Lianying Miao

Commun. Math. Res., 33 (2017), pp. 318-326.

Published online: 2019-11

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  • Abstract

A signed (res. signed total) Roman dominating function, SRDF (res. STRDF) for short, of a graph $G = (V, E)$ is a function $f : V$ → {$−1, 1, 2$} satisfying the conditions that (i) $\sum\limits_{v∈N[v]}f(v) ≥ 1$ (res. $\sum\limits_{v∈N[v]}f(v) ≥ 1$) for any $v ∈ V$ , where $N[v]$ is the closed neighborhood and $N(v)$ is the neighborhood of $v$, and (ii) every vertex $v$ for which $f(v) = −1$ is adjacent to a vertex $u$ for which $f(u) = 2$. The weight of a SRDF (res. STRDF) is the sum of its function values over all vertices. The signed (res. signed total) Roman domination number of $G$ is the minimum weight among all signed (res. signed total) Roman dominating functions of $G$. In this paper, we compute the exact values of the signed (res. signed total) Roman domination numbers of complete bipartite graphs and wheels.

  • Keywords

signed Roman domination, signed total Roman domination, complete bipartite graph, wheel

  • AMS Subject Headings

05C69

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

zhaoyc69@126.com (Yancai Zhao)

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@Article{CMR-33-318, author = {Yancai and Zhao and zhaoyc69@126.com and 5490 and Department of Basic Science, Wuxi City College of Vocational Technology, Wuxi, Jiangsu, 214153 and Yancai Zhao and Lianying and Miao and and 5501 and College of Mathematics, China University of Mining and Technology, Xuzhou, Jiangsu, 221116 and Lianying Miao}, title = {Signed Roman (Total) Domination Numbers of Complete Bipartite Graphs and Wheels}, journal = {Communications in Mathematical Research }, year = {2019}, volume = {33}, number = {4}, pages = {318--326}, abstract = {

A signed (res. signed total) Roman dominating function, SRDF (res. STRDF) for short, of a graph $G = (V, E)$ is a function $f : V$ → {$−1, 1, 2$} satisfying the conditions that (i) $\sum\limits_{v∈N[v]}f(v) ≥ 1$ (res. $\sum\limits_{v∈N[v]}f(v) ≥ 1$) for any $v ∈ V$ , where $N[v]$ is the closed neighborhood and $N(v)$ is the neighborhood of $v$, and (ii) every vertex $v$ for which $f(v) = −1$ is adjacent to a vertex $u$ for which $f(u) = 2$. The weight of a SRDF (res. STRDF) is the sum of its function values over all vertices. The signed (res. signed total) Roman domination number of $G$ is the minimum weight among all signed (res. signed total) Roman dominating functions of $G$. In this paper, we compute the exact values of the signed (res. signed total) Roman domination numbers of complete bipartite graphs and wheels.

}, issn = {2707-8523}, doi = {https://doi.org/10.13447/j.1674-5647.2017.04.04}, url = {http://global-sci.org/intro/article_detail/cmr/13413.html} }
TY - JOUR T1 - Signed Roman (Total) Domination Numbers of Complete Bipartite Graphs and Wheels AU - Zhao , Yancai AU - Miao , Lianying JO - Communications in Mathematical Research VL - 4 SP - 318 EP - 326 PY - 2019 DA - 2019/11 SN - 33 DO - http://doi.org/10.13447/j.1674-5647.2017.04.04 UR - https://global-sci.org/intro/article_detail/cmr/13413.html KW - signed Roman domination, signed total Roman domination, complete bipartite graph, wheel AB -

A signed (res. signed total) Roman dominating function, SRDF (res. STRDF) for short, of a graph $G = (V, E)$ is a function $f : V$ → {$−1, 1, 2$} satisfying the conditions that (i) $\sum\limits_{v∈N[v]}f(v) ≥ 1$ (res. $\sum\limits_{v∈N[v]}f(v) ≥ 1$) for any $v ∈ V$ , where $N[v]$ is the closed neighborhood and $N(v)$ is the neighborhood of $v$, and (ii) every vertex $v$ for which $f(v) = −1$ is adjacent to a vertex $u$ for which $f(u) = 2$. The weight of a SRDF (res. STRDF) is the sum of its function values over all vertices. The signed (res. signed total) Roman domination number of $G$ is the minimum weight among all signed (res. signed total) Roman dominating functions of $G$. In this paper, we compute the exact values of the signed (res. signed total) Roman domination numbers of complete bipartite graphs and wheels.

Yancai Zhao & Lianying Miao. (2019). Signed Roman (Total) Domination Numbers of Complete Bipartite Graphs and Wheels. Communications in Mathematical Research . 33 (4). 318-326. doi:10.13447/j.1674-5647.2017.04.04
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