scholarly journals The Steiner cycle and path cover problem on interval graphs

2021 ◽  
Author(s):  
Ante Ćustić ◽  
Stefan Lendl
Keyword(s):  
Steiner Tree ◽  
Special Class ◽  
Linear Time ◽  
Hamiltonian Path ◽  
Interval Graphs ◽  
Path Cover ◽  
Fixed Set ◽  
Minimum Number ◽  
Cover Problem ◽  
Linear Time Algorithms

AbstractThe 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.

scholarly journals Computing directed Steiner path covers

2021 ◽  
Author(s):  
Frank Gurski ◽  
Dominique Komander ◽  
Carolin Rehs ◽  
Jochen Rethmann ◽  
Egon Wanke
Keyword(s):  
Linear Time ◽  
Hamiltonian Path ◽  
Time Algorithm ◽  
Integer Programs ◽  
Path Cover ◽  
Hamiltonian Path Problem ◽  
Directed Paths ◽  
Minimum Number ◽  
Cover Problem ◽  
Vertex Disjoint

AbstractIn this article we consider the Directed Steiner Path Cover problem on directed co-graphs. Given a directed graph $$G=(V,E)$$ G = ( V , E ) and a set $$T \subseteq V$$ T ⊆ V of so-called terminal vertices, the problem is to find a minimum number of vertex-disjoint simple directed paths, which contain all terminal vertices and a minimum number of non-terminal vertices (Steiner vertices). The primary minimization criteria is the number of paths. We show how to compute in linear time a minimum Steiner path cover for directed co-graphs. This leads to a linear time computation of an optimal directed Steiner path on directed co-graphs, if it exists. Since the Steiner path problem generalizes the Hamiltonian path problem, our results imply the first linear time algorithm for the directed Hamiltonian path problem on directed co-graphs. We also give binary integer programs for the (directed) Hamiltonian path problem, for the (directed) Steiner path problem, and for the (directed) Steiner path cover problem. These integer programs can be used to minimize change-over times in pick-and-place machines used by companies in electronic industry.

scholarly journals Spanning connectedness and Hamiltonian thickness of graphs and interval graphs

2015 ◽  
Vol Vol. 16 no. 2 (PRIMA 2013) ◽  
Author(s):  
Peng Li ◽  
Yaokun Wu
Keyword(s):  
Linear Time ◽  
Hamiltonian Path ◽  
Interval Graphs ◽  
Spanning Subgraph ◽  
Linear Forest ◽  
Connectedness Property ◽  
Vertex Ordering ◽  
International Audience ◽  
Vertex Disjoint ◽  
Linear Time Algorithms

International audience A spanning connectedness property is one which involves the robust existence of a spanning subgraph which is of some special form, say a Hamiltonian cycle in which a sequence of vertices appear in an arbitrarily given ordering, or a Hamiltonian path in the subgraph obtained by deleting any three vertices, or three internally-vertex-disjoint paths with any given endpoints such that the three paths meet every vertex of the graph and cover the edges of an almost arbitrarily given linear forest of a certain fixed size. Let π = π1 · · · πn be an ordering of the vertices of an n-vertex graph G. For any positive integer k ≤ n − 1, we call π a k-thick Hamiltonian vertex ordering of G provided it holds for all i ∈ {1,. .. , n − 1} that πiπi+1 ∈ E(G) and the number of neighbors of πi among {πi+1,. .. , πn} is at least min{n − i, k}; For any nonnegative integer k, we say that π is a −k-thick Hamiltonian vertex ordering of G provided |{i : πiπi+1 / ∈ E(G), 1 ≤ i ≤ n − 1}| ≤ k + 1. Our main discovery is that the existence of a thick Hamiltonian vertex ordering will guarantee that the graph has various kinds of spanning connectedness properties and that for interval graphs, quite a few seemingly unrelated spanning connectedness properties are equivalent to the existence of a thick Hamiltonian vertex ordering. Due to the connection between Hamiltonian thickness and spanning connectedness properties, we can present several linear time algorithms for associated problems. This paper suggests that much work in graph theory may have a spanning version which deserves further study, and that the Hamiltonian thickness may be a useful concept in understanding many spanning connectedness properties.

Linear algorithm for optimal path cover problem on interval graphs

1990 ◽  
Vol 35 (3) ◽  
pp. 149-153 ◽  
Author(s):  
Srinivasa Rao Arikati ◽  
C. Pandu Rangan
Keyword(s):  
Optimal Path ◽  
Interval Graphs ◽  
Linear Algorithm ◽  
Path Cover ◽  
Cover Problem

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

2013 ◽  
pp. 127-138
Author(s):  
Hajo Broersma ◽  
Jiří Fiala ◽  
Petr A. Golovach ◽  
Tomáš Kaiser ◽  
Daniël Paulusma ◽  
...  
Keyword(s):  
Linear Time ◽  
Interval Graphs ◽  
Linear Time Algorithms

A linear-time algorithm for thek-fixed-endpoint path cover problem on cographs

Networks ◽  
2007 ◽  
Vol 50 (4) ◽  
pp. 231-240 ◽  
Author(s):  
Katerina Asdre ◽  
Stavros D. Nikolopoulos
Keyword(s):  
Linear Time ◽  
Time Algorithm ◽  
Linear Time Algorithm ◽  
Path Cover ◽  
Cover Problem

Linear time algorithms for approximating the facility terminal cover problem

Networks ◽  
2007 ◽  
Vol 50 (1) ◽  
pp. 118-126 ◽  
Author(s):  
Guang Xu ◽  
Yang Yang ◽  
Jinhui Xu
Keyword(s):  
Linear Time ◽  
Cover Problem ◽  
Linear Time Algorithms

scholarly journals Hamiltonian Paths in Some Classes of Grid Graphs

2012 ◽  
Vol 2012 ◽  
pp. 1-17 ◽  
Author(s):  
Fatemeh Keshavarz-Kohjerdi ◽  
Alireza Bagheri
Keyword(s):  
Linear Time ◽  
Hamiltonian Path ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Hamiltonian Paths ◽  
Grid Graphs ◽  
Hamiltonian Path Problem ◽  
Np Complete ◽  
Necessary And Sufficient ◽  
Linear Time Algorithms

The Hamiltonian path problem for general grid graphs is known to be NP-complete. In this paper, we give necessary and sufficient conditions for the existence of Hamiltonian paths inL-alphabet,C-alphabet,F-alphabet, andE-alphabet grid graphs. We also present linear-time algorithms for finding Hamiltonian paths in these graphs.

scholarly journals Closed graphs are proper interval graphs

2014 ◽  
Vol 22 (3) ◽  
pp. 37-44
Author(s):  
Marilena Crupi ◽  
Giancarlo Rinaldo
Keyword(s):  
Linear Time ◽  
Interval Graph ◽  
Simple Graph ◽  
Interval Graphs ◽  
Closed Graph ◽  
Graph Recognition ◽  
Proper Interval Graph ◽  
Linear Time Algorithms

Abstract Let G be a connected simple graph. We prove that G is a closed graph if and only if G is a proper interval graph. As a consequence we obtain that there exist linear-time algorithms for closed graph recognition.

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

2014 ◽  
Vol 79 (4) ◽  
pp. 282-299 ◽  
Author(s):  
Hajo Broersma ◽  
Jiří Fiala ◽  
Petr A. Golovach ◽  
Tomáš Kaiser ◽  
Daniël Paulusma ◽  
...  
Keyword(s):  
Linear Time ◽  
Interval Graphs ◽  
Linear Time Algorithms
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