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Toshimasa Ishii Hiroshi Nagamochi Toshihide Ibaraki 《Journal of Combinatorial Optimization》2000,4(1):35-77
Given an undirected multigraph G = (V, E) and two positive integers and k, we consider the problem of augmenting G by the smallest number of new edges to obtain an -edge-connected and k-vertex-connected multigraph. In this paper, we show that the problem can be solved in Õ(mn2) time for any fixed and k = 3 if an input multigraph G is 2-vertex-connected, where n = |V| and m is the number of pairs of adjacent vertices in G. 相似文献
2.
Dieter?VandenbusscheEmail author George?L.?Nemhauser 《Journal of Combinatorial Optimization》2005,9(4):357-379
We study the polyhedron P(G) defined by the convex hull of 2-edge-connected subgraphs of G where multiple copies of edges may be chosen. We show that each vertex of P(G) is also a vertex of the LP relaxation. Given the close relationship with the Graphical Traveling Salesman problem (GTSP), we examine how polyhedral results for GTSP can be modified and applied to P(G). We characterize graphs for which P(G) is integral and study how this relates to a similar result for GTSP. In addition, we show how one can modify some classes of valid inequalities for GTSP and produce new valid inequalities and facets for P(G). 相似文献
3.
Hiroshi Nagamochi Takashi Shiraki Toshihide Ibaraki 《Journal of Combinatorial Optimization》2001,5(2):175-212
Given a finite set V and a set function
, we consider the problem of constructing an undirected multigraph G = (V,E) such that the cut function
together has value at least 2 for all non-empty and proper subsets of V. If f is intersecting submodular and posi-modular, and satisfies the tripartite inequality, then we show that such a multigraph G with the minimum number of edges can be found in
time, where
is the time to compute the value of f(X) for a subset
. 相似文献
4.
For a multigraph G = (V, E), let s V be a designated vertex which has an even degree, and let
G
(V – s) denote min{c
G(X) | Ø X V – s}, where c
G(X) denotes the size of cut X. Splitting two adjacent edges (s, u) and (s, v) means deleting these edges and adding a new edge (u, v). For an integer k, splitting two edges e
1 and e
2 incident to s is called (k, s)-feasible if G(V – s) k holds in the resulting graph G. In this paper, we prove that, for a planar graph G and an even k or k = 3 with k
G
(V – s), there exists a complete (k, s)-feasible splitting at s such that the resulting graph G is still planar, and present an O(n
3 log n) time algorithm for finding such a splitting, where n = |V|. However, for every odd k 5, there is a planar graph G with a vertex s which has no complete (k, s)-feasible and planarity-preserving splitting. As an application of this result, we show that for an outerplanar graph G and an even integer k the problem of optimally augmenting G to a k-edge-connected planar graph can be solved in O(n
3 log n) time. 相似文献
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