Complexity and algorithms for the connected vertex cover problem in 4-regular graphs

In the connected vertex cover (CVC) problem, we are given a connected graph G and required to find a vertex cover set C with minimum cardinality such that the induced subgraph G[C] is connected. In this paper, we restrict our attention to the CVC problem in 4-regular graphs. We proved that the CVC problem is still NP-hard for 4-regular graphs and gave a lower bound for the problem. Moreover, we proposed two approximation algorithms for CVC problem with approximation ratio 32 and 43+O(1n), respectively.

[1]  David Manlove,et al.  Vertex and Edge Covers with Clustering Properties: Complexity and Algorithms , 2009, ACiD.

[2]  Esther M. Arkin,et al.  Approximating the Tree and Tour Covers of a Graph , 1993, Inf. Process. Lett..

[3]  Carla D. Savage,et al.  Depth-First Search and the Vertex Cover Problem , 1982, Inf. Process. Lett..

[4]  Weili Wu,et al.  PTAS for connected vertex cover in unit disk graphs , 2009, Theor. Comput. Sci..

[5]  Toshihiro Fujito,et al.  A 2-approximation NC algorithm for connected vertex cover and tree cover , 2004, Inf. Process. Lett..

[6]  Stefan Richter,et al.  Enumerate and Expand: New Runtime Bounds for Vertex Cover Variants , 2006, COCOON.

[7]  K. Onaga,et al.  Vertex covers and connected vertex covers in 3-connected graphs , 1991, 1991., IEEE International Sympoisum on Circuits and Systems.

[8]  David S. Johnson,et al.  The Rectilinear Steiner Problem is NP-Complete , 1977 .

[9]  David S. Johnson,et al.  Some Simplified NP-Complete Graph Problems , 1976, Theor. Comput. Sci..

[10]  T Hemalatha,et al.  Connected Vertex Cover in 2-Connected Planar Graph with Maximum Degree 4 is NP-complete , 2007 .

[11]  Jérôme Monnot,et al.  Complexity and approximation results for the connected vertex cover problem in graphs and hypergraphs , 2007, J. Discrete Algorithms.

[12]  Yoji Kajitani,et al.  On the nonseparating independent set problem and feedback set problem for graphs with no vertex degree exceeding three , 1988, Discret. Math..

[13]  Rolf Niedermeier,et al.  Parameterized Complexity of Generalized Vertex Cover Problems , 2005, WADS.

[14]  Stefan Richter,et al.  Enumerate and Expand: Improved Algorithms for Connected Vertex Cover and Tree Cover , 2006, Theory of Computing Systems.