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On maximum $$P_3$$ P 3 -packing in claw-free subcubic graphs

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  • Wenying Xi

    (Southeast University)

  • Wensong Lin

    (Southeast University)

Abstract

Let $$P_3$$ P 3 denote the path on three vertices. A $$P_3$$ P 3 -packing of a given graph G is a collection of vertex-disjoint subgraphs of G in which each subgraph is isomorphic to $$P_3$$ P 3 . The maximum $$P_3$$ P 3 -packing problem is to find a $$P_3$$ P 3 -packing of a given graph G which contains the maximum number of vertices of G. The perfect $$P_3$$ P 3 -packing problem is to decide whether a graph G has a $$P_3$$ P 3 -packing that covers all vertices of G. Kelmans (Discrete Appl Math 159:112–127, 2011) proposed the following problem: Is the $$P_3$$ P 3 -packing problem NP-hard in the class of claw-free graphs? In this paper, we solve the problem by proving that the perfect $$P_3$$ P 3 -packing problem in claw-free cubic planar graphs is NP-complete. In addition, we show that for any connected claw-free cubic graph (resp. (2, 3)-regular graph, subcubic graph) G with $$v(G)\ge 14$$ v ( G ) ≥ 14 (resp. $$v(G)\ge 8$$ v ( G ) ≥ 8 , $$v(G)\ge 3$$ v ( G ) ≥ 3 ), the maximum $$P_3$$ P 3 -packing of G covers at least $$\lceil \frac{9v(G)-6}{10} \rceil $$ ⌈ 9 v ( G ) - 6 10 ⌉ (resp. $$\lceil \frac{6v(G)-6}{7} \rceil $$ ⌈ 6 v ( G ) - 6 7 ⌉ , $$\lceil \frac{3v(G)-6}{4} \rceil $$ ⌈ 3 v ( G ) - 6 4 ⌉ ) vertices, where v(G) denotes the order of G, and the bound is sharp. The proofs imply polynomial-time algorithms.

Suggested Citation

  • Wenying Xi & Wensong Lin, 2021. "On maximum $$P_3$$ P 3 -packing in claw-free subcubic graphs," Journal of Combinatorial Optimization, Springer, vol. 41(3), pages 694-709, April.
    • Handle: RePEc:spr:jcomop:v:41:y:2021:i:3:d:10.1007_s10878-021-00708-2
    DOI: 10.1007/s10878-021-00708-2
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    References listed on IDEAS

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    1. Qilong Feng & Jianxin Wang & Shaohua Li & Jianer Chen, 2015. "Randomized parameterized algorithms for $$P_2$$ P 2 -Packing and Co-Path Packing problems," Journal of Combinatorial Optimization, Springer, vol. 29(1), pages 125-140, January.
    2. Henning Fernau & Daniel Raible, 2009. "A parameterized perspective on packing paths of length two," Journal of Combinatorial Optimization, Springer, vol. 18(4), pages 319-341, November.
    3. Maw-Shang Chang & Li-Hsuan Chen & Ling-Ju Hung, 2016. "An $$O^{*}(1.4366^n)$$ O ∗ ( 1 . 4366 n ) -time exact algorithm for maximum $$P_2$$ P 2 -packing in cubic graphs," Journal of Combinatorial Optimization, Springer, vol. 32(2), pages 594-607, August.
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    Cited by:

    1. Wenying Xi & Wensong Lin, 2024. "The maximum 3-star packing problem in claw-free cubic graphs," Journal of Combinatorial Optimization, Springer, vol. 47(5), pages 1-16, July.

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