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

Wenying Xi and Wensong Lin ()
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Wenying Xi: Southeast University
Wensong Lin: Southeast University

Journal of Combinatorial Optimization, 2021, vol. 41, issue 3, No 6, 694-709

Abstract: 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.

Keywords: Cubic graph; (2; 3)-Regular graph; Subcubic graph; Claw-free graph; $$P_3$$ P 3 -packing; Perfect $$P_3$$ P 3 -packing; 05C15 (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1007/s10878-021-00708-2

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