How to Find Long Paths Efficiently
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TLDR
In this paper, the complexity of finding a long path in directed or undirected graphs was studied and an algorithm for finding a longest path with complexity O(K! · |V| · |E|) was proposed.Abstract:
We study the complexity of finding long paths in directed or undirected graphs, Given a graph G =(V, E) and a number k our algorithm decides within time O(K! · |V| · |E|) for all u,v ɛ V Whether there exists some path of length k firm u to v. The complexity of this algorithm has to be compared with 0(|V| k−1 · |E|) Which is the worst case behaviour of the algorithms described up to now in the literature, We get similar results for the problems of finding a longest path, a cycle of length k or a longest cycle, respectively. Our approach is based on the idea of representing certain families of sets by subfamilies of small cardiality. We also discuss the border lines of this idea.read more
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References
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Proceedings ArticleDOI
Finding a minimum circuit in a graph
Alon Itai,Michael Rodeh +1 more
TL;DR: Finding minimum circuits in graphs and digraphs is discussed and an algorithm to find an almost minimum circuit is presented and an alternative method is to reduce the problem of finding a minimum circuit to that of finding an auxiliary graph.
Journal ArticleDOI
A Problem in Graph Theory
TL;DR: In this article, it is shown that with the addition of any new edge a compIete k-graph is formed, where each edge joins a vertex to itself and at most one edge joins any two vertices.
Journal ArticleDOI
A polynomial algorithm for maximum weighted vertex packings on graphs without long odd cycles
TL;DR: An O(n2K+1) algorithm for the maximum weighted vertex packing problem for graphs inG ≥ 1 and an algorithm for piecing together maximum weighted packings on blocks to find maximum weightedPackings on graphs that contain more than one block are presented.
Journal ArticleDOI
Relative lengths of paths and cycles in k-connected graphs
TL;DR: It is shown that if G contains a path L of length l then G also contains a cycle of length at least ((2k − 4)(3k −4) l) l, which together cover every edge of L at least 2k2 − 6k + 4 times.