Finding a longest path in a complete multipartite digraph

A digraph obtained by replacing each edge of a complete $m$-partite graph with an arc or a pair of mutually opposite arcs with the same end vertices is called a complete $m$-partite digraph. An $O ( n^3 )$ algorithm for finding a longest path in a complete $m$-partite $( m \geq 2 )$ digraph with $n$...

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Bibliographic Details
Published inSIAM journal on discrete mathematics Vol. 6; no. 2; pp. 270 - 273
Main Author GUTIN, G
Format Journal Article
LanguageEnglish
Published Philadelphia, PA Society for Industrial and Applied Mathematics 01.05.1993
Subjects
Algorithms
Combinatorics
Combinatorics. Ordered structures
Exact sciences and technology
Graph theory
Graphs
Mathematics
Sciences and techniques of general use
Multipartite graph
Longest path
Complete graph
Directed graph
Hamiltonian path
Search algorithm
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Summary:A digraph obtained by replacing each edge of a complete $m$-partite graph with an arc or a pair of mutually opposite arcs with the same end vertices is called a complete $m$-partite digraph. An $O ( n^3 )$ algorithm for finding a longest path in a complete $m$-partite $( m \geq 2 )$ digraph with $n$ vertices is described in this paper. The algorithm requires time $O( n^{2.5} )$ in case of testing only the existence of a Hamiltonian path and finding it if one exists. It is simpler than the algorithm of Manoussakis and Tuza [SIAM J. Discrete Math., 3 (1990), pp. 537-543], which works only for $m = 2$. The algorithm implies a simple characterization of complete $m$-partite digraphs having Hamiltonian paths that was obtained for the first time in Gutin [Kibernetica (Kiev), 4 (1985), pp. 124-125] for $m = 2$ and in Gutin [Kibernetica (Kiev), 1(1988), pp. 107-108] for $ m \geq 2 $.
ISSN:0895-4801
1095-7146
DOI:10.1137/0406020