| Summary: | 碩士 === 國立中正大學 === 資訊工程研究所 === 82 === Let G=(V,E) be an undirected graph and let [s_i,t_i],1 .ltoreq.
i .ltoreq. k, be k pairs of vertices of G. The vertex-disjoint
paths problem is to find k paths P_1,...,P_k such that P_i
connects s_i and t_i for 1 .ltoreq. i .ltoreq. k and any two
paths P_i and P_j are vertex disjoint. this problem is NP-
complete even for planar graphs. Robertson and Seymour proved
that when k is a fixed integer this problem becomes polynomial.
The edge-disjoint paths problem is to find k paths P_1,..., P_k
such that P_i connects s_i and t_i for 1 .ltoreq. i .ltoreq. k
and any two paths P_i and P_j are edge disjoint. In this
thesis, a polynomial algorithm is given for the vertex-disjoint
paths problem on block graphs. This implies that the edge-
disjoint paths problemis NP-complete on all graphs containing
the class of complete graphs. The results are interesting in
two aspects :(1) Very few problem are NP-complete on complete
graphs and (2) The edge-fisjoint paths problem is much harder
than the vertex-disjoint paths problem on block graphs. We also
prove that the edge-disjoint paths problem, restricting {s_i,
t_i} .neq. {s_j,t_j} for all 1 .ltoreq. i<j .ltoreq. k and (s_i,
t_i) .notin. E for all 1 .ltoreq. i .ltoreq. k, is NP-complete
on block graphs and split graphs.
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