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We discuss the recently introduced concept of k-in-out graphs, and provide aconstruction for k-in-out graphs for any positive integer k. We derive a lowerbound for the number of vertices of a k-in-out graph for any positive integerk, and demonstrate that our construction meets this bound in all cases. Foreven k, we also prove our construction is optimal with respect to the number ofedges, and results in a planar graph. Among the possible uses of in-out graphs,they can convert the generalized traveling salesman problem to the asymmetrictraveling salesman problem, avoiding the "big M" issue present in most otherconversions. We give constraints satisfied by all in-out graphs to assistcutting-plane algorithms in solving instances of traveling salesman problemwhich contain in-out graphs.

A construction for directed in-out subgraphs of optimal size