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

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- Received:
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DOI: 10.1007/s00373-009-0851-y

- Cite this article as:
- Narayanaswamy, N.S., Sadagopan, N. & Chandran, L.S. Graphs and Combinatorics (2009) 25: 557. doi:10.1007/s00373-009-0851-y

## 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 2*k*-connected partial 2*k*-trees.

### Keywords

ConnectivityContractionContractible edgesPartial*k*-trees