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

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

Z. N. Berberler, M. Çerezci

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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.

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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。

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