An algorithm for (n-3)-connectivity augmentation problem: Jump system approach

  • Authors:

  • Kristóf Bérczi;Yusuke Kobayashi

  • Affiliations:

  • Department of Operations Research, Eötvös Loránd University, Budapest, Hungary;Department of Mathematical Informatics, Graduate School of Information Science and Technology, University of Tokyo, Tokyo, Japan

  • Venue:
  • Journal of Combinatorial Theory Series B
  • Year:
  • 2012

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Abstract

We consider the problem of making a given (k-1)-connected graph k-connected by adding a minimum number of new edges, which we call the k-connectivity augmentation problem. In this paper, we deal with the problem when k=n-3 where n is the number of vertices of the input graph. By considering the complement graph, the (n-3)-connectivity augmentation problem can be reduced to the problem of finding a maximum square-free 2-matching in a simple graph with maximum degree at most three. We give a polynomial-time algorithm to find a maximum square-free 2-matching in a simple graph with maximum degree at most three, which yields a polynomial-time algorithm for the (n-3)-connectivity augmentation problem. Our algorithm is based on the fact that the square-free 2-matchings are endowed with a matroid structure called a jump system. We also show that the weighted (n-3)-connectivity augmentation problem can be solved in polynomial time if the weights are induced by a function on the vertex set, whereas the problem is NP-hard for general weights.