On computing longest paths in small graph classes
Ryuhei Uehara,Yushi Uno +1 more
TLDR
It is shown that the longest path problem can be solved efficiently for some tree-like graph classes by this approach, and two new graph classes that have natural interval representations are proposed.Abstract:
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.read more
Citations
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Transmission expansion planning with re-design
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Polynomial fixed-parameter algorithms : a case study for longest path on interval graphs.
TL;DR: This work studies the design of fixed-parameter algorithms for problems already known to be solvable in polynomial time, and shows how to solve Longest Path on Interval Graphs, parameterized by vertex deletion number k to proper interval graphs, in O(k9n) time.
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A linear-time algorithm for the longest path problem in rectangular grid graphs
TL;DR: This paper gives a linear-time algorithm for finding a longest path between any two given vertices in a rectangular grid graph.
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The 1-Fixed-Endpoint Path Cover Problem is Polynomial on Interval Graphs
TL;DR: In this paper, a polynomial-time algorithm for the 1-fixed-endpoint path cover problem on interval graphs was proposed, which is a generalization of the 1HP problem.
Posted Content
The 1-fixed-endpoint Path Cover Problem is Polynomial on Interval Graph
TL;DR: It is shown that the 1PC problem can be solved in polynomial time on the class of interval graphs, generalizing the 1HP problem which has been proved to be NP-complete even for small classes of graphs.
References
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TL;DR: This new Annals edition continues to convey the message that intersection graph models are a necessary and important tool for solving real-world problems and remains a stepping stone from which the reader may embark on one of many fascinating research trails.