On the Structure of Contractible Edges in k connected Partial k trees
 N. S. Narayanaswamy,
 N. Sadagopan,
 L. Sunil Chandran
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Contraction of an edge e merges its end points into a new single vertex, and each neighbor of one of the end points of e is a neighbor of the new vertex. An edge in a kconnected graph is contractible if its contraction does not result in a graph with lesser connectivity; otherwise the edge is called noncontractible. In this paper, we present results on the structure of contractible edges in ktrees and kconnected partial ktrees. Firstly, we show that an edge e in a ktree is contractible if and only if e belongs to exactly one (k + 1) clique. We use this characterization to show that the graph formed by contractible edges is a 2connected graph. We also show that there are at least V(G) + k − 2 contractible edges in a ktree. Secondly, we show that if an edge e in a partial ktree is contractible then e is contractible in any ktree which contains the partial ktree as an edge subgraph. We also construct a class of contraction critical 2kconnected partial 2ktrees.
 Title
 On the Structure of Contractible Edges in kconnected Partial ktrees
 Journal

Graphs and Combinatorics
Volume 25, Issue 4 , pp 557569
 Cover Date
 200911
 DOI
 10.1007/s003730090851y
 Print ISSN
 09110119
 Online ISSN
 14355914
 Publisher
 Springer Japan
 Additional Links
 Topics
 Keywords

 Connectivity
 Contraction
 Contractible edges
 Partial ktrees
 Industry Sectors
 Authors

 N. S. Narayanaswamy ^{(1)}
 N. Sadagopan ^{(1)}
 L. Sunil Chandran ^{(2)}
 Author Affiliations

 1. Department of Computer Science and Engineering, Indian Institute of Technology, Chennai, 600036, India
 2. Computer Science and Automation, Indian Institute of Science, Bangalore, 560012, India