Studies in 3-connected graphs: A decomposition theory and a decomposition-based optimization algorithm
Abstract
Cunningham and Edmonds (1980) have proved that a 2-connected graph G has a unique minimal decomposition into graphs each of which is either 3-connected, a bond or a polygon. They define the notion of a good split, and first prove that G has a unique decomposition into graphs, none of which have a good split, and second prove that the graphs that do not have a good split are precisely 3-connected graphs, bonds and polygons. This thesis provides an analogue of the first result above for 3-connected graphs, and an analogue of the second for minimally 3-connected graphs. Following the basic strategy of Cunningham and Edmonds, an appropriate notion of good split is defined. The first main result if that if G is a 3-connected graph, then G has a unique minimal decomposition into graphs, none of which have a good split. The second main result is that the minimally 3-connected graphs that do not have a good split are precisely cyclically 4-connected graphs, twirls ($K\sb{3,n}$ for some $n$ $\geq$ 3) and wheels. From this it is shown that if G is a minimally 3-connected graph, then G has a unique minimal decomposition into graphs, each of which is cyclically 4-connected, a twirl or a wheel. Hopcroft and Tarjan (1973) gave a polynomial-time algorithm for dividing a 2-connected graph into 3-connected components. The third main result of this thesis is a pair of polynomial-time algorithms for finding the unique minimal decomposition of a minimally 3-connected graph into graphs, each of which is cyclically 4-connected, a twirl or a wheel. Finally, a polynomial-time algorithm is given for the minimum-weight cycle problem on graphs that may be decomposed into twirls and wheels.
Degree
Ph.D.
Advisors
Wagner, Purdue University.
Subject Area
Operations research
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