Exploring the influence of graph operations on zero forcing sets

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics Pub Date : 2025-04-05 DOI:10.1016/j.disc.2025.114516

Krishna Menon , Anurag Singh

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引用次数: 0

Abstract

Zero forcing in graphs is a coloring process where a vertex colored blue can force its unique uncolored neighbor to be colored blue. A zero forcing set is a set of initially blue vertices capable of eventually coloring all vertices of the graph. In this paper, we focus on the numbers

z (G; i)

, which is the number of zero forcing sets of size i of the graph G. These numbers were initially studied by Boyer et al. [5] where they conjectured that for any graph G on n vertices,

z (G; i) \leq z (P_{n}; i)

for all

i \geq 1

where

P_{n}

is the path graph on n vertices. The main aim of this paper is to show that several classes of graphs, including outerplanar graphs and threshold graphs, satisfy this conjecture. We do this by studying various graph operations and examining how they affect the number of zero forcing sets.

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探讨图运算对零强迫集的影响

图中的零强制是一种着色过程，其中一个着色为蓝色的顶点可以强制其唯一的未着色的邻居着色为蓝色。零强迫集是一组初始为蓝色的顶点，能够最终为图形的所有顶点上色。在本文中，我们关注的是数字z(G;i)，即图G中大小为i的零强迫集的个数。这些数字最初是由Boyer等人[[5]]研究的，他们推测对于任意n个顶点上的图G，对于所有i≥1，z(G;i)≤z(Pn;i)，其中Pn为n个顶点上的路径图。本文的主要目的是证明几类图，包括外平面图和阈值图，满足这个猜想。我们通过研究各种图操作并检查它们如何影响零强迫集的数量来做到这一点。

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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来源期刊

Discrete Mathematics 数学-数学

CiteScore

1.50

自引率

12.50%

发文量

424

审稿时长

6 months

期刊介绍： Discrete Mathematics provides a common forum for significant research in many areas of discrete mathematics and combinatorics. Among the fields covered by Discrete Mathematics are graph and hypergraph theory, enumeration, coding theory, block designs, the combinatorics of partially ordered sets, extremal set theory, matroid theory, algebraic combinatorics, discrete geometry, matrices, and discrete probability theory. Items in the journal include research articles (Contributions or Notes, depending on length) and survey/expository articles (Perspectives). Efforts are made to process the submission of Notes (short articles) quickly. The Perspectives section features expository articles accessible to a broad audience that cast new light or present unifying points of view on well-known or insufficiently-known topics.