Abstract
A join in a graph is a set F of edges such that for every circuit C, |C ∩ F| ≤ |C ∖ F|. We study the problem of finding a connected join covering a given subset of vertices of the graph, that is a Steiner tree which is a join at the same time. This turns out to contain the question of finding a T-join of minimum cardinality (or weight) which is, in addition, connected. This last problem is mentioned to be open in a survey of Frank [7], and is motivated by its link to integral packings of T-cuts: if a minimum T-join F is connected, then there exists an integral packing of T-cuts of cardinality |F|.
The problems we deal with are closely related to some known NP- complete problems: deciding the existence of a connected T-join; finding the minimum cardinality of a connected T-join; the Steiner tree problem; subgraph isomorphism. We also explore some of these connections.
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Sebő, A., Tannier, E. (2001). connected Joins in Graphs. In: Aardal, K., Gerards, B. (eds) Integer Programming and Combinatorial Optimization. IPCO 2001. Lecture Notes in Computer Science, vol 2081. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45535-3_30
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DOI: https://doi.org/10.1007/3-540-45535-3_30
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