Abstract
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 k-connected graph is contractible if its contraction does not result in a graph with lesser connectivity; otherwise the edge is called non-contractible. In this paper, we present results on the structure of contractible edges in k-trees and k-connected partial k-trees. Firstly, we show that an edge e in a k-tree 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 2-connected graph. We also show that there are at least |V(G)| + k − 2 contractible edges in a k-tree. Secondly, we show that if an edge e in a partial k-tree is contractible then e is contractible in any k-tree which contains the partial k-tree as an edge subgraph. We also construct a class of contraction critical 2k-connected partial 2k-trees.
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Narayanaswamy, N.S., Sadagopan, N. & Chandran, L.S. On the Structure of Contractible Edges in k-connected Partial k-trees. Graphs and Combinatorics 25, 557–569 (2009). https://doi.org/10.1007/s00373-009-0851-y
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DOI: https://doi.org/10.1007/s00373-009-0851-y