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Orientable Burning Number of Graphs

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WALCOM: Algorithms and Computation (WALCOM 2024)

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Abstract

In this paper, we introduce the problem of finding an orientation of a given undirected graph that maximizes the burning number of the resulting directed graph. We show that the problem is polynomial-time solvable on Kőnig–Egerváry graphs (and thus on bipartite graphs) and that an almost optimal solution can be computed in polynomial time for perfect graphs. On the other hand, we show that the problem is NP-hard in general and W[1]-hard parameterized by the target burning number. The hardness results are complemented by several fixed-parameter tractable results parameterized by structural parameters. Our main result in this direction shows that the problem is fixed-parameter tractable parameterized by cluster vertex deletion number plus clique number (and thus also by vertex cover number).

Partially supported by JSPS KAKENHI Grant Numbers JP18H04091, JP20H05793, JP21K11752, JP22H00513, JP23KJ1066. The full version of this paper is available at https://arxiv.org/abs/2311.13132.

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Notes

  1. 1.

    In a directed graph, a vertex dominates itself and its out-neighbors.

  2. 2.

    There is another way for handling orientation by using a variant of MSO\(_{2}\) defined for directed graphs, where we can fix an arbitrary orientation first (without using a k-coloring) and then represent reversed edges by an edge set. See e.g., [18].

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Authors and Affiliations

  1. Normandie Univ, UNICAEN, ENSICAEN, CNRS, GREYC, 14000, Caen, France

    Julien Courtiel, Paul Dorbec & Romain Lecoq

  2. Nagoya University, Nagoya, Japan

    Tatsuya Gima & Yota Otachi

  3. JSPS Research Fellow, Tokyo, Japan

    Tatsuya Gima

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  1. Julien Courtiel
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Correspondence to Yota Otachi .

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  1. Japan Advanced Institute of Science and Technology, Nomi, Japan

    Ryuhei Uehara

  2. Iwate University, Morioka, Japan

    Katsuhisa Yamanaka

  3. National Taiwan University, Taipei, Taiwan

    Hsu-Chun Yen

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Courtiel, J., Dorbec, P., Gima, T., Lecoq, R., Otachi, Y. (2024). Orientable Burning Number of Graphs. In: Uehara, R., Yamanaka, K., Yen, HC. (eds) WALCOM: Algorithms and Computation. WALCOM 2024. Lecture Notes in Computer Science, vol 14549. Springer, Singapore. https://doi.org/10.1007/978-981-97-0566-5_27

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