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

令\（P_3 \）表示三个顶点上的路径。甲\（P_3 \）分装给定图的ģ是顶点不相交子图的集合ģ，其中每个子图同构于\（P_3 \） 。最大\（P_3 \）分装问题是要找到一个\（P_3 \）分装在给定图形的ģ其中包含的顶点的最大数目ģ。完美的\（P_3 \）-打包问题是确定图G是否具有覆盖G的所有顶点的\（P_3 \）-打包。Kelmans（Discrete Appl Math 159：112–127，2011）提出了以下问题：无爪图类中的\（P_3 \）-打包问题NP-hard吗？本文通过证明无爪三次平面图中的完美\（P_3 \）-堆积问题是NP-完全的来解决该问题的。此外，我们还表明，对于任何连接的无爪立方曲线图（分别（2,3）规则图形，图表subcubic）ģ与\（V（G）\ GE 14 \） （相应地，\（V（G ）\ GE 8 \） ，\（v（G）\ GE 3 \） ），最大\（P_3 \）的分装ģ至少覆盖\（\ lceil \压裂{9V（G）-6} {10 } \ rceil \）（分别为\（\ lceil \ frac {6v（G）-6} {7} \ rceil \），\（\ lceil \压裂{3V（G）-6} {4} \ rceil \） ）的顶点，其中，v（g ^）表示的顺序ģ，并且结合是尖锐。证明暗示多项式时间算法。