The 1-fixed-endpoint Path Cover Problem is Polynomial on Interval Graph (original) (raw)

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.

A sufficient condition for the unpaired k-disjoint path coverability of interval graphs

The Journal of Supercomputing

t k }, in a graph G, an unpaired k-disjoint path cover joining S and T is a set of pairwise vertex-disjoint paths {P 1 ,. .. , P k } that altogether cover every vertex of the graph, in which P i is a path from source s i to some sink t j. In this paper, we prove that if the scattering number, sc(G), of an interval graph G of order n ≥ 2k is less than or equal to −k, there exists an unpaired k-disjoint path cover joining S and T in G for any possible configurations of source and sink sets S and T of size k each. The bound sc(G) ≤ −k is tight; moreover, the proof directly leads to a quadratic algorithm for building an unpaired k-disjoint path cover.

The Steiner cycle and path cover problem on interval graphs

2021

The Steiner path problem is a common generalization of the Steiner tree and the Hamiltonian path problem, in which we have to decide if for a given graph there exists a path visiting a fixed set of terminals. In the Steiner cycle problem we look for a cycle visiting all terminals instead of a path. The Steiner path cover problem is an optimization variant of the Steiner path problem generalizing the path cover problem, in which one has to cover all terminals with a minimum number of paths. We study those problems for the special class of interval graphs. We present linear time algorithms for both the Steiner path cover problem and the Steiner cycle problem on interval graphs given as endpoint sorted lists. The main contribution is a lemma showing that backward steps to non-Steiner intervals are never necessary. Furthermore, we show how to integrate this modification to the deferred-query technique of Chang et al. to obtain the linear running times.

Characterization of interval graphs that are unpaired 2-disjoint path coverable

Theoretical Computer Science, 2020

Given disjoint source and sink sets, S = {s 1 ,. .. , s k } and T = {t 1 ,. .. , t k }, in a graph G, an unpaired k-disjoint path cover joining S and T is a set of pairwise vertex-disjoint paths {P 1 ,. .. , P k } that altogether cover every vertex of the graph, in which P i is a path from source s i to some sink t j. In terms of a generalized scattering number, named an r-scattering number, we characterize interval graphs that have an unpaired 2-disjoint path cover joining S and T for any possible configurations of source and sink sets S and T of size 2 each. Also, it is shown that the r-scattering number of an interval graph can be computed in polynomial time.

Assistance and Interdiction Problems on Interval Graphs

2021

We introduce a novel framework of graph modifications specific to interval graphs. We study interdiction problems with respect to these graph modifications. Given a list of original intervals, each interval has a replacement interval such that either the replacement contains the original, or the original contains the replacement. The interdictor is allowed to replace up to k original intervals with their replacements. Using this framework we also study the contrary of interdiction problems which we call assistance problems. We study these problems for the independence number, the clique number, shortest paths, and the scattering number. We obtain polynomial time algorithms for most of the studied problems. Via easy reductions, it follows that on interval graphs, the most vital nodes problem with respect to shortest path, independence number and Hamiltonicity can be solved in polynomial time.

An Optimal Algorithm to Solve 2-Neighbourhood Covering Problem on Interval Graphs

International Journal of Computer Mathematics, 2002

Let G (V, E) be a simple graph and k be a ®xed positive integer. A vertex w is said to be a kneighbourhood-cover of an edge (u, v) if d(u, w) k and d(v, w) k. A set C V is called a kneighbourhood-covering set if every edge in E is k-neighbourhood-covered by some vertices of C. This problem is NP-complete for general graphs even it remains NP-complete for chordal graphs. Using dynamic programming technique, an O(n) time algorithm is designed to solve minimum 2-neighbourhood-covering problem on interval graphs. A data structure called interval tree is used to solve this problem.

Recognizing graphs with fixed interval number is NP-complete

Discrete Applied Mathematics, 1984

A t-interval representation of a graph expresses it as the intersection graph of a family of subsets of the real line. Each vertex is assigned a set consisting of at most t disjoint closed intervals, in such a way that vertices are adjacent if and only if some interval for one intersects some interval for the other. The interval number i(G) of a graph G is the smallest number t such that G has a t-representation. We prove that, for any fixed value of t with t ~2, determining whether i(G)5 t is NP-complete.

On the Classes of Interval Graphs of Limited Nesting and Count of Lengths

Algorithmica

In 1969, Roberts introduced proper and unit interval graphs and proved that these classes are equal. Natural generalizations of unit interval graphs called k-length interval graphs were considered in which the number of different lengths of intervals is limited by k. Even after decades of research, no insight into their structure is known and the complexity of recognition is open even for k = 2. We propose generalizations of proper interval graphs called k-nested interval graphs in which there are no chains of k + 1 intervals nested in each other. It is easy to see that k-nested interval graphs are a superclass of k-length interval graphs. We give a linear-time recognition algorithm for k-nested interval graphs. This algorithm adds a missing piece to Gajarský et al. [FOCS 2015] to show that testing FO properties on interval graphs is FPT with respect to the nesting k and the length of the formula, while the problem is W[2]-hard when parameterized just by the length of the formula. Further, we show that a generalization of recognition called partial representation extension is polynomial-time solvable for k-nested interval graphs, while it is NP-hard for k-length interval graphs, even when k = 2.