Characterization of interval graphs that are unpaired 2-disjoint path coverable (original) (raw)

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 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...

Linear-Time Algorithms for Scattering Number and Hamilton-Connectivity of Interval Graphs

Lecture Notes in Computer Science, 2013

Hung and Chang showed that for all k ≥ 1 an interval graph has a path cover of size at most k if and only if its scattering number is at most k. They also showed that an interval graph has a Hamilton cycle if and only if its scattering number is at most 0. We complete this characterization by proving that for all k ≤ −1 an interval graph is −(k + 1)-Hamilton-connected if and only if its scattering number is at most k. We also give an O(m + n) time algorithm for computing the scattering number of an interval graph with n vertices an m edges, which improves the O(n 4) time bound of Kratsch, Kloks and Müller. As a consequence of our two results the maximum k for which an interval graph is k-Hamilton-connected can be computed in O(m + n) 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.

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.

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.

On Restrictions of Balanced 2-Interval Graphs

WG'07, 2007

The class of 2-interval graphs has been introduced for modelling scheduling and allocation problems, and more recently for specific bioinformatics problems. Some of those applications imply restrictions on the 2-interval graphs, and justify the introduction of a hierarchy of subclasses of 2-interval graphs that generalize line graphs: balanced 2-interval graphs, unit 2-interval graphs, and (x,x)-interval graphs. We provide instances that show that all inclusions are strict. We extend the NP-completeness proof of recognizing 2-interval graphs to the recognition of balanced 2-interval graphs. Finally we give hints on the complexity of unit 2-interval graphs recognition, by studying relationships with other graph classes: proper circular-arc, quasi-line graphs, K 1,5-free graphs, ...

General-demand disjoint path covers in a graph with faulty elements

International Journal of Computer Mathematics, 2012

A k-disjoint path cover of a graph is a set of k internally vertex-disjoint paths which cover the vertex set with k paths and each of which runs between a source and a sink. Given that each source and sink v is associated with an integer-valued demand d(v) ≥ 1, we are concerned with general-demand k-disjoint path cover in which every source and sink v is contained in the d(v) paths. In this paper, we present a reduction of a general-demand disjoint path cover problem to an unpaired many-to-many disjoint path cover problem, and obtain some results on disjoint path covers of restricted HL-graphs and proper interval graphs with faulty vertices and/or edges.

Remarks on the Interval Number of Graphs

Acta Cybernetica

The interval number of a graph G is the least natural number t such that G is the intersection graph of sets, each of which is the union of at most t intervals. Here, we propose a family of representations for the graph G, which yield the well-known upper bound ⌈1 2(d+1)⌉, where d is the maximum degree of G. The extremal graphs for even d are also described, and the upper bound on the interval number in terms of the number of edges of G is improved.