The 1-Fixed-Endpoint Path Cover Problem is Polynomial on Interval Graphs (original) (raw)
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The 1-fixed-endpoint Path Cover Problem is Polynomial on Interval Graph
Computing Research Repository - CORR, 2008
We consider a variant of the path cover problem, namely, the kkk-fixed-endpoint path cover problem, or kPC for short, on interval graphs. Given a graph GGG and a subset mathcalT\mathcal{T}mathcalT of kkk vertices of V(G)V(G)V(G), a kkk-fixed-endpoint path cover of GGG with respect to mathcalT\mathcal{T}mathcalT is a set of vertex-disjoint paths mathcalP\mathcal{P}mathcalP that covers the vertices of GGG such that the kkk vertices of mathcalT\mathcal{T}mathcalT are all endpoints of the paths in mathcalP\mathcal{P}mathcalP. The kPC problem is to find a kkk-fixed-endpoint path cover of GGG of minimum cardinality; note that, if mathcalT\mathcal{T}mathcalT is empty the stated problem coincides with the classical path cover problem. In this paper, we study the 1-fixed-endpoint path cover problem on interval graphs, or 1PC for short, generalizing the 1HP problem which has been proved to be NP-complete even for small classes of graphs. Motivated by a work of Damaschke, where he left both 1HP and 2HP problems open for the class of interval graphs, we show that the 1PC problem can...
A polynomial solution to the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" display="inline" overflow="scroll">mml:mik-fixed-endpoint path cover problem on proper interval graphs
Theoretical Computer Science, 2010
We study a variant of the path cover problem, namely, the k-fixed-endpoint path cover problem, or kPC for short. Given a graph G and a subset T of k vertices of V (G), a k-fixed-endpoint path cover of G with respect to T is a set of vertex-disjoint paths P that covers the vertices of G such that the k vertices of T are all endpoints of the paths in P. The kPC problem is to find a k-fixed-endpoint path cover of G of minimum cardinality; note that, if T is empty (or, equivalently, k = 0), the stated problem coincides with the classical path cover problem. The kPC problem generalizes some path cover related problems, such as the 1HP and 2HP problems, which have been proved to be NP-complete. Note that, the complexity status of both 1HP and 2HP problems on interval graphs remains an open question [9]. In this paper, we show that the kPC problem can be solved in linear time on the class of proper interval graphs, that is, in O(n + m) time on a proper interval graph on n vertices and m edges. The proposed algorithm is simple, requires linear space, and also enables us to solve the 1HP and 2HP problems, on proper interval graphs within the same time and space complexity.
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