 
Summary: Complexity of Finding a Join of Maximum
Weight ?
A. A. Ageev 1
Sobolev Institute of Mathematics, 630090, Novosibirsk, Russia
Abstract
A subset of edges J E(G) in a undirected graph G is called a join if at most
half the edges of each cycle of G are contained in J . In this paper we consider the
problem of nding a join of maximum weight: given a graph G and an edge weighting
c : E(G) ! R, nd a join of maximum weight. We show that the problem is NP
hard even in the case of 0; 1weights, which answers in the negative a question of A.
Frank. We also show that in the case of seriesparallel graphs and arbitrary weights
the problem can be solved in time O(n 3 ), where n is the number of vertices in G.
Key words: Graph; Join; NPhard problem; Seriesparallel graph;
Polynomialtime algorithm
1 Introduction
In this paper a graph stands for an undirected multigraph without loops.
Let G be a graph. A subset of edges J E(G) is called a join of G, if at
most half the edges of each cycle in G belong to J . We consider the problem of
nding a join of maximum weight that can be stated as follows: given a graph
G and an edge weighting c : E(G) ! R, nd a join of G with maximum
