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

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

05C69

zhaoyc69@126.com (Yancai Zhao)

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@Article{CMR-33-318, author = {Zhao , Yancai and Miao , Lianying}, 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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