DMGT

ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

https://doi.org/10.7151/dmgt

Discussiones Mathematicae Graph Theory

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Discussiones Mathematicae Graph Theory

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Discussiones Mathematicae Graph Theory 23(1) (2003) 5-21
DOI: 10.7151/dmgt.1182

BALANCED PROBLEMS ON GRAPHS WITH CATEGORIZATION OF EDGES

Stefan Berezný

Air Force Academy of M.R. Stefánik, Košice
Rampová 7, 041 21 Košice, Slovakia

Vladimír Lacko

Institute of Mathematics
University of P.J. Safárik, Košice
Jesenná 5, 041 54 Košice, Slovakia
e-mail: lacko@science.upjs.sk

Abstract

Suppose a graph G = (V,E) with edge weights w(e) and edges partitioned into disjoint categories S1,... ,Sp is given. We consider optimization problems P on G defined by a family of feasible sets D(G) and the following objective function:
L5(D)
=

max
1 ≤ i ≤ p 


max
e ∈ Si∩D 
w(e)−
min
e ∈ Si∩D 
w(e)

For an arbitrary number of categories we show that the L5-perfect matching, L5-a-b path, L5-spanning tree problems and L5-Hamilton cycle (on a Halin graph) problem are NP-complete.

We also summarize polynomiality results concerning above objective functions for arbitrary and for fixed number of categories. 

Keywords: algorithms on graphs, categorization of edges, NP-completeness.

2000 Mathematics Subject Classification: 05C85, 90C27, 68Q17.

References

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Received 18 December 2000
Revised 6 May 2002