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Characterizing 3-uniform linear extremal hypergraphs on feedback vertex number

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Abstract

Let \(H=(V,E)\) be a hypergraph with vertex set V and edge set E. \(S\subseteq V\) is a feedback vertex set (FVS) of H if \(H\setminus S\) has no cycle and \(\tau _c(H)\) denote the minimum cardinality of a FVS of H. Chen et al. [IWOCA,2016] has proven if H is a linear 3-uniform hypergraph with m edges, then \(\tau _c(H)\le m/3\). In this paper, we furthermore characterize all the extremal hypergraphs with \(\tau _c(H)= m/3\) holds. This result has a direct application to Tuza’s conjecture.

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Acknowledgements

The authors are very indebted to Professor Xujin Chen, Professor Xiaodong Hu and the anonymous referee for their invaluable comments and suggestions. This research work is supported by National Natural Science Foundation of China under Grant Nos.11901605, 11901292, 12101069, the disciplinary funding of Central University of Finance and Economics, the Emerging Interdisciplinary Project of CUFE, the Fundamental Research Funds for the Central Universities and Innovation Foundation of BUPT for Youth (500422309).

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Authors and Affiliations

  1. School of Science, Beijing University of Posts and Telecommunications, Beijing, 100876, China

    Zhongzheng Tang

  2. Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, 211106, China

    Yucong Tang

  3. Key Laboratory of Mathematical Modelling and High Performance Computing of Air Vehicles (NUAA), MIIT, Nanjing, 211106, China

    Yucong Tang

  4. School of Statistics and Mathematics, Central University of Finance and Economics, Beijing, 100081, China

    Zhuo Diao

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  1. Zhongzheng Tang
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  2. Yucong Tang
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  3. Zhuo Diao
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Correspondence to Zhuo Diao.

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A preliminary version of this paper appeared in Proceedings of the 15th International Conference on Combinatorial Optimization and Applications, pp. \(687-700, 2021\)

Appendix: Codes for testing Case 3 in Lemmas

Appendix: Codes for testing Case 3 in Lemmas

In this appendix, we show all the Python codes for testing the Case 3 of Lemma 5 and Lemma 6.

First, we define the data structures of edges, vertices and undirected linear hypergraphs. The data structure of hypergraphs supports the operations of adding and deleting vertices or edges.

figure b

The second function aims to check whether a hypergraph contains cycles. The idea of this algorithm is breadth-first search(BFS) on edges.

figure c

The third function is to check whether two hypergraphs are isomorphic. The method is to enumerate all the permutations of vertices restricted by the vertex degree sequences of hypergraphs. Notice that the worst running time of this algorithm is exponential, so it is only suitable for small or medium scale hypergraphs.

figure d

The fourth algorithm is to deal with Case 3 in Lemma 5 and Lemma 6. First, generate all possible hypergraphs by adding a new 3-degree vertex to a non-trivial component. Next, check whether each new hypergraph is extremal. Finally, take advantage of the isomorphic algorithm to remove duplicate hypergraphs.

figure e

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Tang, Z., Tang, Y. & Diao, Z. Characterizing 3-uniform linear extremal hypergraphs on feedback vertex number. J Comb Optim 44, 3310–3330 (2022). https://doi.org/10.1007/s10878-022-00893-8

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