The 2-Terminal-Set Path Cover Problem and Its Polynomial Solution on Cographs

Abstract

In this paper we study a generalization of the path cover problem, namely, the 2-terminal-set path cover problem, or 2TPC for short. Given a graph G and two disjoint subsets $\mathcal{T}^1$ and $\mathcal{T}^2$ of V(G), a 2-terminal-set path cover of G with respect to $\mathcal{T}^1$ and $\mathcal{T}^2$ is a set of vertex-disjoint paths $\mathcal{P}$ that covers the vertices of G such that the vertices of $\mathcal{T}^1$ and $\mathcal{T}^2$ are all endpoints of the paths in $\mathcal{P}$ and all the paths with both endpoints in $\mathcal{T}^1 \cup \mathcal{T}^2$ have one endpoint in $\mathcal{T}^1$ and the other in $\mathcal{T}^2$. The 2TPC problem is to find a 2-terminal-set path cover of G of minimum cardinality; note that, if $\mathcal{T}^1 \cup \mathcal{T}^2$ is empty, the stated problem coincides with the classical path cover problem. The 2TPC problem generalizes some path cover related problems, such as the 1HP and 2HP problems, which have been proved to be NP-complete even for small classes of graphs. We show that the 2TPC problem can be solved in linear time on the class of cographs. The proposed linear-time algorithm is simple, requires linear space, and also enables us to solve the 1HP and 2HP problems on cographs within the same time and space complexity.

The research Project is co-funded by the European Union - European Social Fund (ESF) & National Sources, in the framework of the program ”Pythagoras II” of the 3rd Community Support Framework of the Hellenic Ministry of Education.