scholarly journals Optimal Hamiltonian completions and path covers for trees, and a reduction to maximum flow

2002 ◽  
Vol 44 (2) ◽  
pp. 193-204 ◽  
Author(s):  
D. S. Franzblau ◽  
A. Raychaudhuri
Keyword(s):  
Linear Time ◽  
Optimal Path ◽  
Hamiltonian Path ◽  
Maximum Flow ◽  
Time Algorithm ◽  
Frequency Assignment ◽  
Minimum Size ◽  
Linear Time Algorithm ◽  
Path Cover ◽  
Vertex Disjoint

A minimum Hamiltonian completion of a graph G is a minimum-size set of edges that, when added to G, guarantee a Hamiltonian path. Finding a Hamiltonian completion has applications to frequency assignment as well as distributed computing. If the new edges are deleted from the Hamiltonian path, one is left with a minimum path cover, a minimum-size set of vertex-disjoint paths that cover the vertices of G. For arbitrary graphs, constructing a minimum Hamiltonian completion or path cover is clearly NP-hard, but there exists a linear-time algorithm for trees. In this paper we first give a description and proof of correctness for this linear-time algorithm that is simpler and more intuitive than those given previously. We show that the algorithm extends also to unicyclic graphs. We then give a new method for finding an optimal path cover or Hamiltonian completion for a tree that uses a reduction to a maximum flow problem. In addition, we show how to extend the reduction to construct, if possible, a covering of the vertices of a bipartite graph with vertex-disjoint cycles, that is, a 2-factor.

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 A Linear Time Algorithm for a Variant of the MAX CUT Problem in Series Parallel Graphs

2017 ◽  
Vol 2017 ◽  
pp. 1-4 ◽  
Author(s):  
Brahim Chaourar
Keyword(s):  
Linear Time ◽  
Maximum Flow ◽  
Time Algorithm ◽  
Minimum Cut ◽  
Linear Time Algorithm ◽  
Induced Subgraphs ◽  
Positive Weight ◽  
Max Cut Problem ◽  
In Series ◽  
Minimum Cut Problem

Given a graph G=V,E, a connected sides cut U,V\U or δU is the set of edges of E linking all vertices of U to all vertices of V\U such that the induced subgraphs GU and GV\U are connected. Given a positive weight function w defined on E, the maximum connected sides cut problem (MAX CS CUT) is to find a connected sides cut Ω such that wΩ is maximum. MAX CS CUT is NP-hard. In this paper, we give a linear time algorithm to solve MAX CS CUT for series parallel graphs. We deduce a linear time algorithm for the minimum cut problem in the same class of graphs without computing the maximum flow.

LINEAR-TIME ALGORITHMS FOR DISJOINT TWO-FACE PATHS PROBLEMS IN PLANAR GRAPHS

1996 ◽  
Vol 07 (02) ◽  
pp. 95-110 ◽  
Author(s):  
HEIKE RIPPHAUSEN-LIPA ◽  
DOROTHEA WAGNER ◽  
KARSTEN WEIHE
Keyword(s):  
Planar Graphs ◽  
Linear Time ◽  
Time Algorithm ◽  
Disjoint Paths ◽  
Linear Time Algorithm ◽  
Vertex Disjoint ◽  
Linear Time Algorithms

In this paper we present a linear-time algorithm for the vertex-disjoint Two-Face Paths Problem in planar graphs, i.e., the problem of finding k vertex-disjoint paths between pairs of terminals which lie on two face boundaries. The algorithm is based on the idea of finding rightmost paths with a certain property in planar graphs. Using this method, a linear-time algorithm for finding vertex-disjoint paths of a prescribed homotopy is derived. Moreover, the algorithm is modified to solve the more general linkage problem in linear time, as well.

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

scholarly journals ON COMPUTING LONGEST PATHS IN SMALL GRAPH CLASSES

2007 ◽  
Vol 18 (05) ◽  
pp. 911-930 ◽  
Author(s):  
RYUHEI UEHARA ◽  
YUSHI UNO
Keyword(s):  
Linear Time ◽  
Hamiltonian Path ◽  
Time Algorithm ◽  
Linear Time Algorithm ◽  
Graph Classes ◽  
Longest Path ◽  
Longest Path Problem ◽  
Longest Paths ◽  
Interval Representations ◽  
The One

The longest path problem is the one that finds a longest path in a given graph. While the graph classes in which the Hamiltonian path problem can be solved efficiently are widely investigated, few graph classes are known to be solved efficiently for the longest path problem. Among those, for trees, a simple linear time algorithm for the longest path problem is known. We first generalize the algorithm, and show that the longest path problem can be solved efficiently for some tree-like graph classes by this approach. We next propose two new graph classes that have natural interval representations, and show that the longest path problem can be solved efficiently on these classes.

scholarly journals Total Roman {3}-Domination: The Complexity and Linear-Time Algorithm for Trees

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 293
Author(s):  
Xinyue Liu ◽  
Huiqin Jiang ◽  
Pu Wu ◽  
Zehui Shao
Keyword(s):  
Linear Time ◽  
Minimum Weight ◽  
Domination Number ◽  
Time Algorithm ◽  
Simple Graph ◽  
Linear Time Algorithm ◽  
Domination Problem ◽  
Np Complete ◽  
Chordal Bipartite Graphs ◽  
Dominating Function

For a simple graph G=(V,E) with no isolated vertices, a total Roman {3}-dominating function(TR3DF) on G is a function f:V(G)→{0,1,2,3} having the property that (i) ∑w∈N(v)f(w)≥3 if f(v)=0; (ii) ∑w∈N(v)f(w)≥2 if f(v)=1; and (iii) every vertex v with f(v)≠0 has a neighbor u with f(u)≠0 for every vertex v∈V(G). The weight of a TR3DF f is the sum f(V)=∑v∈V(G)f(v) and the minimum weight of a total Roman {3}-dominating function on G is called the total Roman {3}-domination number denoted by γt{R3}(G). In this paper, we show that the total Roman {3}-domination problem is NP-complete for planar graphs and chordal bipartite graphs. Finally, we present a linear-time algorithm to compute the value of γt{R3} for trees.

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