A Note on Contractible Edges in Chordal Graphs

TL;DR

This paper characterizes contractible edges in chordal graphs by analyzing their properties through tree decompositions and minimal vertex separators, providing insights into preserving connectivity during edge contractions.

Contribution

It introduces a characterization of contractible edges in chordal graphs based on tree decompositions and minimal vertex separators, advancing understanding of graph connectivity preservation.

Findings

Identifies conditions for edge contractibility in chordal graphs

Uses properties of tree decompositions to analyze edge contraction

Provides a framework for preserving connectivity in graph modifications

Abstract

Contraction of an edge merges its end points into a new vertex which is adjacent to each neighbor of the end points of the edge. An edge in a $k$-connected graph is {\em contractible} if its contraction does not result in a graph of lower connectivity. We characterize contractible edges in chordal graphs using properties of tree decompositions with respect to minimal vertex separators.