A covering problem that is easy for trees but $\mathrm{NP}$-complete for trivalent graphs

Abstract

By definition, a P2-graph $\Gamma$ is an undirected graph in which every vertex is contained in a path of length two. For such a graph, $\mathrm{pc}\left(\Gamma \right)$ denotes the minimum number of paths of length two that cover all $n$ vertices of $\Gamma$. We prove that $\lceil n/3\rceil \le \mathrm{pc}\left(\Gamma \right)\le \lfloor n/2\rfloor$ and show that these upper and lower bounds are tight. Furthermore we show that every connected P2-graph $\Gamma$ contains a spanning tree $T$ such that $\mathrm{pc}\left(\Gamma \right)=\mathrm{pc}\left(T\right)$. We present a linear time algorithm that produces optimal 2-path covers for trees. This is contrasted by the result that the decision problem $\mathrm{pc}\left(\Gamma \right)\stackrel{?}{=}n/3$ is $\mathrm{NP}$-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.