Algorithmic aspects of {Pk}-isolation in graphs and extremal graphs for a {P3}-isolation bound

IF 0.7 4区计算机科学 Q4 COMPUTER SCIENCE, INFORMATION SYSTEMS

Information Processing Letters Pub Date : 2024-07-14 DOI:10.1016/j.ipl.2024.106521

Jie Chen , Yi-Ping Liang , Cai-Xia Wang , Shou-Jun Xu

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引用次数: 0

Abstract

A subset S of the vertex set of a graph G is an $F$ -isolating set of G if $G - N [S]$ does not contain a copy of a member of $F$ as a subgraph, where $F$ is a family of connected graphs and $N [S]$ is the closed neighborhood of S. The $F$ -isolation number of G is the minimum cardinality of an $F$ -isolating set of G, denoted by $ι (G, F)$ . Given a graph G, ${P_{k}}$ -ISOLATION asks for the size of a smallest ${P_{k}}$ -isolating set of G for a fixed positive integer k, where $P_{k}$ is a path of order k. In this paper, we first show that the decision version of ${P_{k}}$ -ISOLATION is NP-complete for chordal graphs and planar graphs. Secondly, we propose a linear time algorithm to compute a smallest ${P_{k}}$ -isolating set of a tree. Finally, we solve the problem of characterizing the connected graphs G with $ι (G, {P_{3}}) = \frac{2}{7} | V (G) |$ , proposed by Zhang and Wu [Discrete Appl. Math. 304 (2021) 365-374].

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图中{Pk}隔离的算法方面以及{P3}隔离约束的极值图

如果 G-N[S] 不包含作为子图的 F 成员的副本，则图 G 的顶点集的子集 S 是 G 的 F 隔离集，其中 F 是连通图族，N[S] 是 S 的封闭邻域。G 的 F 隔离数是 G 的 F 隔离集的最小心性，用 ι(G,F) 表示。给定一个图 G，{Pk}-ISOLATION 会求解在固定正整数 k 条件下 G 的最小 {Pk} 隔离集的大小，其中 Pk 是阶数为 k 的路径。在本文中，我们首先证明对于弦图和平面图，{Pk}-ISOLATION 的判定版本是 NP-完全的。其次，我们提出了一种计算树的最小 {Pk} 隔离集的线性时间算法。最后，我们解决了张和吴提出的具有 ι(G,{P3})=27|V(G)| 的连通图 G 的特征问题[离散应用数学 304 (2021) 365-374]。

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来源期刊

Information Processing Letters 工程技术-计算机：信息系统

CiteScore

1.80

自引率

0.00%

发文量

审稿时长

7.3 months

期刊介绍： Information Processing Letters invites submission of original research articles that focus on fundamental aspects of information processing and computing. This naturally includes work in the broadly understood field of theoretical computer science; although papers in all areas of scientific inquiry will be given consideration, provided that they describe research contributions credibly motivated by applications to computing and involve rigorous methodology. High quality experimental papers that address topics of sufficiently broad interest may also be considered. Since its inception in 1971, Information Processing Letters has served as a forum for timely dissemination of short, concise and focused research contributions. Continuing with this tradition, and to expedite the reviewing process, manuscripts are generally limited in length to nine pages when they appear in print.