ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

Discussiones Mathematicae Graph Theory

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Discussiones Mathematicae Graph Theory

Article in press


F.X. Liu, B. Wu and J.X. Meng


Arbitrarily partitionable $\{2K_2, C_4\}$-free graphs


Discussiones Mathematicae Graph Theory

Received: 2018-12-02, Revised: 2019-11-30, Accepted: 2019-11-30,


A graph $G=(V,E)$ of order $n$ is said to be arbitrarily partitionable if for each sequence $\lambda=(\lambda_1, \lambda_2, \dots, \lambda_p)$ of positive integers with $\lambda_1+\cdots+\lambda_p=n$, there exists a partition $(V_1, V_2, \dots, V_p)$ of the vertex set $V$ such that $V_i$ induces a connected subgraph of order $\lambda_i$ in $G$ for each $i\in \{1,2,\dots,p\}$. In this paper, we show that a threshold graph is arbitrarily partitionable if and only if it admits a perfect matching or a near perfect matching. We also give a necessary and sufficient condition for a $\{2K_2, C_4\}$-free graph being arbitrarily partitionable, as an extension for a result of Broersma, Kratsch and Woeginger [Fully decomposable split graphs, European J. Combin. 34 (2013) 567–575] on split graphs.


arbitrarily partitionable graphs, arbitrarily vertex decomposable, threshold graphs, $\{2K_2, C_4\}$-free graphs