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Cut and pendant vertices and the number of connected induced subgraphs of a graph

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Abstract

A vertex whose removal in a graph G increases the number of components of G is called a cut vertex. For all n, c, we determine the maximum number of connected induced subgraphs in a connected graph with order n and c cut vertices, and also characterise those graphs attaining the bound. Moreover, we show that the cycle has the smallest number of connected induced subgraphs among all cut vertex-free connected graphs. The general case \(c>0\) remains an open task. We also characterise the extremal graph structures given both order and number of pendant vertices, and establish the corresponding formulas for the number of connected induced subgraphs. The ‘minimal’ graph in this case is a tree, thus coincides with the structure that was given by Li and Wang (Electron J Comb 19(4):P48, 2012).

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Acknowledgements

The author is pleased to acknowledge a discussion with Stephan Wagner.

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Authors and Affiliations

  1. Department of Mathematics and Applied Mathematics, University of Johannesburg, P.O. Box 524, Auckland Park, Johannesburg, 2006, South Africa

    Audace A. V. Dossou-Olory

  2. Institut National de l’Eau (INE), Université d’Abomey-Calavi (UAC), Abomey-Calavi, Benin

    Audace A. V. Dossou-Olory

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Partially supported by the National Research Foundation of South Africa, Grant 118521.

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Dossou-Olory, A.A.V. Cut and pendant vertices and the number of connected induced subgraphs of a graph. European Journal of Mathematics 7, 766–792 (2021). https://doi.org/10.1007/s40879-020-00443-8

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