http://hdl.handle.net/1880/45328 2013-06-19T11:22:59Z http://hdl.handle.net/1880/45610 Title: HALIN GRAPHS AND THE TRAVELLING SALESMAN PROBLEM Authors: Cornuejols, G.; Naddef, D.; Pulleyblank, W.R. Abstract: A Halin graph $H = T cup C$ is obtained by imbedding a tree $T$ having no degree two nodes in the plane, and then adding a cycle $C$ to join the leaves of $T$ in such a way that the resulting graph is planar. These graphs are edge minimal 3-connected, hamiltonian, and in general have large numbers of hamilton cycles. We show that for arbitrary real edge costs the travelling salesman problem can be polynomially solved for such a graph, and we give an explicit linear description of the travelling salesman polytope (the convex hull of the incidence vectors of the hamilton cycles) for such a graph. 1981-11-01T00:00:00Z http://hdl.handle.net/1880/45609 Title: A LINEAR PROGRAMMING RELAXATION OF THE NODE PACKING PROBLEM OR 2-BICRITICALGRAPHS AND NODE COVERS Authors: Pulleyblank, William R. Abstract: The problem of finding a minimum cardinality set of nodes in a graph which meet every edge is of considerable theoretical as well as practical interest. Because of the difficulty of this problem, a linear relaxation of an integer programming model is sometimes used as a heuristic. In fact Nemhauser and Trotter showed that any variables which receive integer values in an optimal solution to the relaxation can retain the same values in an optimal solution to the integer program. We define 2-bicritical graphs and give several characterizations of them. One characterization is that they are precisely the graphs for which an optimal solution to the linear relaxation will have no integer valued variables. Then we show that almost all graphs are 2-bicritical, and hence the linear relaxation almost never helps. 1978-03-01T00:00:00Z http://hdl.handle.net/1880/45608 Title: MIXED DOUBLES TABLE TENNIS TOURNAMENTS Authors: Pulleyblank, W.R. Abstract: No Abstract 1976-01-01T00:00:00Z