Covering a Graph with Nontrivial Vertex-Disjoint Paths: Existence and Optimization

Abstract

Let G be a connected graph and $\mathcal{P}$ be a set of pairwise vertex-disjoint paths in G. We say that $\mathcal{P}$ is a path cover if every vertex of G belongs to a path in $\mathcal{P}$. In the minimum path cover problem, one wishes to find a path cover of minimum cardinality. In this problem, known to be ${\textsc {NP}}$-hard, the set $\mathcal{P}$ may contain trivial (single-vertex) paths. We study the problem of finding a path cover composed only of nontrivial paths. First, we show that the corresponding existence problem can be reduced to a matching problem on a bipartite graph via the Edmonds-Gallai Decomposition. This reduction gives, in polynomial time, a certificate for both the yes-answer and the no-answer. When trivial paths are forbidden, for the feasible instances, one may consider either minimizing or maximizing the number of paths in the path cover. We show that the maximization problem has a close relation with the maximum matchings of a graph, and can be solved in polynomial time. For the minimization problem on feasible instances, we show that its computational complexity is equivalent to the minimum path cover problem. We also show a linear-time algorithm on (edge-weighted) trees.

Research supported by CNPq (Proc. 456792/2014-7, 306464/2016-0), FAPESP (Proc. 2015/11937-9), CAPES (235671298-48), MaCLinC Proj. NUMEC/USP, Brazil.