A covering problem that is easy for trees but I-complete for trivalent graphs

By definition, a P2-graph Γ is an undirected graph in which every vertex is contained in a path of length two. For such a graph, pc(Γ) denotes the minimum number of paths of length two that cover all n vertices of Γ. We prove that ⌈n/3⌉ ≤ pc(Γ) ≤ ⌊n/2⌋ and show that these upper and lower bounds are tight. Furthermore we show that every connected P2-graph Γ contains a spanning tree T such that pc(Γ) = pc(T ). We present a linear time algorithm that produces optimal 2-path covers for trees. This is contrasted by the result that the decision problem pc(Γ) ? = n/3 is NP-complete for trivalent graphs. This graph theoretical problem originates from the task of searching a large database of biological molecules such as the Protein Data Bank (PDB) by content.