More Aspects of Arbitrarily Partitionable Graphs

IF 0.5 4区数学 Q3 MATHEMATICS

Discussiones Mathematicae Graph Theory Pub Date : 2022-07-12 DOI:10.7151/DMGT.2343

Julien Bensmail, Binlong Li

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引用次数: 3

Abstract

Abstract A graph G of order n is arbitrarily partitionable (AP) if, for every sequence (n1, . . ., np) partitioning n, there is a partition (V1, . . ., ,Vp) of V (G) such that G[Vi] is a connected ni-graph for i = 1, . . ., p. The property of being AP is related to other well-known graph notions, such as perfect matchings and Hamiltonian cycles, with which it shares several properties. This work is dedicated to studying two aspects behind AP graphs. On the one hand, we consider algorithmic aspects of AP graphs, which received some attention in previous works. We first establish the NP-hardness of the problem of partitioning a graph into connected subgraphs following a given sequence, for various new graph classes of interest. We then prove that the problem of deciding whether a graph is AP is in NP for several classes of graphs, confirming a conjecture of Barth and Fournier for these. On the other hand, we consider the weakening to APness of su cient conditions for Hamiltonicity. While previous works have suggested that such conditions can sometimes indeed be weakened, we here point out cases where this is not true. This is done by considering conditions for Hamiltonicity involving squares of graphs, and claw- and net-free graphs.

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任意可分图的更多方面

摘要n阶图G是任意可分图（AP），如果对于每个划分n的序列（n1，…，np），存在V（G）的划分（V1，…，Vp），使得G[Vi]是i＝1。，p。AP的性质与其他众所周知的图概念有关，如完全匹配和哈密顿循环，它与这些概念有几个共同的性质。这项工作致力于研究AP图背后的两个方面。一方面，我们考虑了AP图的算法方面，这在以前的工作中受到了一些关注。我们首先针对各种感兴趣的新图类，建立了将图划分为遵循给定序列的连通子图的问题的NP硬度。然后，我们证明了判定图是否为AP的问题对于几类图是NP的，证实了Barth和Fournier对这些图的猜想。另一方面，我们考虑了哈密顿性的充分条件对AP性的削弱。虽然之前的工作表明，这种条件有时确实会被削弱，但我们在这里指出了一些情况，但事实并非如此。这是通过考虑图的平方、爪和无网图的哈密顿性的条件来实现的。

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

Discussiones Mathematicae Graph Theory MATHEMATICS-

CiteScore

2.20

自引率

0.00%

发文量

审稿时长

53 weeks

期刊介绍： The Discussiones Mathematicae Graph Theory publishes high-quality refereed original papers. Occasionally, very authoritative expository survey articles and notes of exceptional value can be published. The journal is mainly devoted to the following topics in Graph Theory: colourings, partitions (general colourings), hereditary properties, independence and domination, structures in graphs (sets, paths, cycles, etc.), local properties, products of graphs as well as graph algorithms related to these topics.