Unique Response Roman Domination Versus 2-Packing Differential in Complementary Prisms

IF 0.6 4区数学 Q3 MATHEMATICS

Mathematical Notes Pub Date : 2024-07-15 DOI:10.1134/s0001434624050237

Z. N. Berberler, M. Çerezci

{"title":"Unique Response Roman Domination Versus 2-Packing Differential in Complementary Prisms","authors":"Z. N. Berberler, M. Çerezci","doi":"10.1134/s0001434624050237","DOIUrl":null,"url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3> Let \$G = (V,E)\$ be a graph of order \$n\$. For \$S \\subseteq V(G)\$, the set \$N_e(S)\$ is defined as the external neighborhood of \$S\$ such that all vertices in \$V(G)\\backslash S\$ have at least one neighbor in \$S\$. The differential of \$S\$ is defined to be \$\\partial(S)=|N_e(S)|-|S|\$, and the 2-packing differential of a graph is defined as $$\\partial_{2p}(G) =\\max\\{\\partial(S)\\colon S \\subseteq V(G) \\text{ is a 2-packing}\\}.$$ A function \$f\\colon V(G) \\to \\{0,1,2\\}\$ with the sets \$V_0,V_1,V_2\$, where $$V_i =\\{v\\in V(G)\\colon f(v) = i\\},\\qquad i \\in \\{0,1,2\\},$$ is a unique response Roman dominating function if \$x \\in V_0 \$ implies that \$| N( x ) \\cap V_2 | = 1\$ and \$x \\in V_1 \\cup V_2 \$ implies that \$N( x ) \\cap V_2 = \\emptyset\$. The unique response Roman domination number of \$G\$, denoted by \$\\mu_R(G)\$, is the minimum weight among all unique response Roman dominating functions on \$G\$. Let \$\\bar{G}\$ be the complement of a graph \$G\$. The complementary prism \$G\\bar {G}\$ of \$G\$ is the graph formed from the disjoint union of \$G\$ and \$\\bar {G}\$ by adding the edges of a perfect matching between the respective vertices of \$G\$ and \$\\bar {G}\$. The present paper deals with the computation of the 2-packing differential and the unique response Roman domination of the complementary prisms \$G\\bar {G}\$ by the use of a proven Gallai-type theorem. Particular attention is given to the complementary prims of special types of graphs. Furthermore, the graphs \$G\$ such that \$\\partial_{2p} ( G\\bar G)\$ and \$\\mu _R(G\\bar G)\$ are small are characterized. ","PeriodicalId":18294,"journal":{"name":"Mathematical Notes","volume":"48 1","pages":""},"PeriodicalIF":0.6000,"publicationDate":"2024-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematical Notes","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1134/s0001434624050237","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}

引用次数: 0

Abstract

Let $G = (V,E)$ be a graph of order $n$. For $S \subseteq V(G)$, the set $N_e(S)$ is defined as the external neighborhood of $S$ such that all vertices in $V(G)\backslash S$ have at least one neighbor in $S$. The differential of $S$ is defined to be $\partial(S)=|N_e(S)|-|S|$, and the 2-packing differential of a graph is defined as

$$\partial_{2p}(G) =\max\{\partial(S)\colon S \subseteq V(G) \text{ is a 2-packing}\}.$$

A function $f\colon V(G) \to \{0,1,2\}$ with the sets $V_0,V_1,V_2$, where

$$V_i =\{v\in V(G)\colon f(v) = i\},\qquad i \in \{0,1,2\},$$

is a unique response Roman dominating function if $x \in V_0 $ implies that $| N( x ) \cap V_2 | = 1$ and $x \in V_1 \cup V_2 $ implies that $N( x ) \cap V_2 = \emptyset$. The unique response Roman domination number of $G$, denoted by $\mu_R(G)$, is the minimum weight among all unique response Roman dominating functions on $G$. Let $\bar{G}$ be the complement of a graph $G$. The complementary prism $G\bar {G}$ of $G$ is the graph formed from the disjoint union of $G$ and $\bar {G}$ by adding the edges of a perfect matching between the respective vertices of $G$ and $\bar {G}$. The present paper deals with the computation of the 2-packing differential and the unique response Roman domination of the complementary prisms $G\bar {G}$ by the use of a proven Gallai-type theorem. Particular attention is given to the complementary prims of special types of graphs. Furthermore, the graphs $G$ such that $\partial_{2p} ( G\bar G)$ and $\mu _R(G\bar G)$ are small are characterized.

查看原文本刊更多论文

互补棱镜中的独特响应罗马统治与 2-Packing 差异

Abstract Let $G = (V,E)$ be a graph of order $n$.对于 $S \subseteq V(G)$, 集合 $N_e(S)$ 被定义为 $S$ 的外部邻域，使得 $V(G)\backslash S$ 中的所有顶点在 $S$ 中至少有一个邻域。$S$的微分被定义为$\partial(S)=|N_e(S)|-|S|$，图的 2-packing 微分被定义为 $$\partial_{2p}(G) =\max\{partial(S)\colon S \subseteq V(G) \text{ is a 2-packing}\}.一个函数（f/colon V(G)/to/{0,1,2}/）的集合是（V_0,V_1,V_2），其中$$V_i =\{v\in V(G)\colon f(v) = i\},\qquad i \in \{0,1,2}、如果 $x \in V_0 $ 意味着 $| N( x ) \cap V_2 | = 1$ 并且 $x \in V_1 \cup V_2 $ 意味着 $N( x ) \cap V_2 = \emptyset$，那么 $$就是唯一的响应罗马支配函数。G\) 的唯一响应罗马支配数用 $\mu_R(G)$ 表示，它是$G$ 上所有唯一响应罗马支配函数中的最小权值。让 $\bar{G}$ 成为图 $G$ 的补集。G\ 的互补棱图是由$G$和$\bar {G}$的互不相交的联合图通过添加$G$和$\bar {G}$各自顶点之间完美匹配的边而形成的图。本文通过使用已被证明的伽来定理，讨论了互补棱柱 $G\bar {G}$的 2-packing differential 和 unique response Roman domination 的计算。我们特别关注了特殊类型图的补捯。此外，还描述了使$\partial_{2p} ( G\bar G)$和$\mu _R(G\bar G)$都很小的图$G$的特征。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文求助全文

来源期刊

Mathematical Notes 数学-数学

CiteScore

0.90

自引率

16.70%

发文量

179

审稿时长

24 months

期刊介绍： Mathematical Notes is a journal that publishes research papers and review articles in modern algebra, geometry and number theory, functional analysis, logic, set and measure theory, topology, probability and stochastics, differential and noncommutative geometry, operator and group theory, asymptotic and approximation methods, mathematical finance, linear and nonlinear equations, ergodic and spectral theory, operator algebras, and other related theoretical fields. It also presents rigorous results in mathematical physics.