We consider a variant of the path cover problem, namely, the k-fixedendpoint path cover problem, or kPC for short, on interval graphs. 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 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 (Discrete Math. 112:49-64, 1993), where he left both 1HP and 2HP problems open for the class of interval graphs, we show that the 1PC problem can be solved in polynomial time on the class of interval graphs. We propose a polynomial-time algorithm for the problem, which also enables us to solve the 1HP problem on interval graphs within the same time and space complexity.
In this paper, we prove that the harmonious coloring problem is NP-complete for connected interval and permutation graphs. Given a simple graph G, a harmonious coloring of G is a proper vertex coloring such that each pair of colors appears together on at most one edge. The harmonious chromatic number is the least integer k for which G admits a harmonious coloring with k colors. Extending previous work on the NP-completeness of the harmonious coloring problem when restricted to the class of disconnected graphs which are simultaneously cographs and interval graphs, we prove that the problem is also NP-complete for connected interval and permutation graphs.
In this paper, we study a variant of the path cover problem, namely, the k -fixed-endpoint path cover problem. Given a graph G and a subset T of k vertices of V (G), a kfixed-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 k -fixed-endpoint path cover problem is to find a k -fixed-endpoint path cover of G of minimum cardinality; note that, if T is empty, that is, k = 0, the stated problem coincides with the classical path cover problem. We show that the k -fixed-endpoint path cover problem can be solved in linear time on the class of cographs. More precisely, we first establish a lower bound on the size of a minimum k -fixed-endpoint path cover of a cograph and prove structural properties for the paths of such a path cover. Then, based on these properties, we describe an algorithm which, for a cograph G on n vertices and m edges, computes a minimum k -fixed-endpoint path cover of G in linear time, that is, in O(n + m) time. The proposed algorithm is simple, requires linear space, and also enables us to solve some path cover related problems, such as the 1HP and 2HP, on cographs within the same time and space complexity.
a b s t r a c tWe 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-fixedendpoint 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 for both 1HP and 2HP problems on interval graphs remains an open question (Damaschke (1993) [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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