@article {IOPORT.06099139,
    author       = {Molina, Robert and McNally, Myles},
    title        = {Randomly $P_k$-decomposable graphs.},
    year         = {2012},
    journal      = {Discrete Mathematics},
    volume       = {312},
    number       = {22},
    issn         = {0012-365X},
    pages        = {3406-3416},
    publisher    = {Elsevier Science B.V. (North-Holland), Amsterdam},
    doi          = {10.1016/j.disc.2012.07.036},
    abstract     = {Summary: A graph $G$ is $H$-decomposable if it can be expressed as an edge-disjoint union of subgraphs, each subgraph isomorphic to $H$. If $G$ has the additional property that every $H$-decomposable subgraph of $G$ is part of an $H$-decomposition of $G$, then $G$ is randomly $H$-decomposable. Using computer assistance, we provide in this paper a characterization of randomly path-decomposable graphs for paths of length 11 or less. We also prove the following two results: { indent=7mm \item{(1)}With one small exception, randomly $P_{k}$-decomposable graphs with a vertex of odd degree do not contain a $P_{k+1}$-subgraph, \item{(2)} When the edges of a $P_{k}$-subgraph are deleted from a connected randomly $P_{k}$-decomposable graph, the resulting graph has at most one nontrivial component. }},
    identifier   = {06099139},
}