A Linear-Time Algorithm for the Terminal Path Cover Problem in Block Graphs

In this paper, we study a variant of the path cover problem, namely, the terminal path cover problem. Given a graph G and a subset T of vertices of G, a terminal path cover of G with respect to T is a set of pairwise vertex-disjoint paths PC that covers the vertices of G such that the vertices of T are all endpoints of the paths in PC. The terminal path cover problem is to find a terminal path cover of G of minimum cardinality; note that, if T is empty, the stated problem coincides with the classical path cover problem. We show that the terminal path cover problem can be solved in linear time on the class of block graphs. More precisely, we first establish a tree structural representation for the class of block graphs. Then, based on the tree structure, we present an algorithm which, for a block graph G on n vertices and m edges, computes a minimum terminal path cover of G in linear time, that is, in O(n+m) time. The proposed algorithm is simple and only requires linear space.

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