On the Structure of Contractible Edges in k-connected Partial k-trees

Narayanaswamy, NS and Sadagopan, N and Chandran, Sunil L (2009) On the Structure of Contractible Edges in k-connected Partial k-trees. In: Graphs and Combinatorics, 25 (4). pp. 557-569.

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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.

Item Type:	Journal Article
Publication:	Graphs and Combinatorics
Publisher:	Springer
Additional Information:	Copyright for this article belongs to Springer.
Keywords:	Connectivity; Contraction; Contractible edges; Partial k-trees
Department/Centre:	Division of Electrical Sciences > Computer Science & Automation
Date Deposited:	05 Jan 2010 11:49
Last Modified:	28 Feb 2019 08:35
URI:	http://eprints.iisc.ac.in/id/eprint/25336

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