Abstract

We introduce and study a partial order on graphs—lift-contractions. A graph H is a lift-contraction of a graph G if H can be obtained from G by a sequence of edge lifts and edge contractions. We give sufficient conditions for a connected graph to contain every n-vertex graph as a lift-contraction and describe the structure of graphs with an excluded lift-contraction.

Highlights

  • All graphs in this paper are undirected, loopless, and without multiple edges

  • We say that a graph H is a lift-contraction of a graph G if H can be obtained from G by a sequence of edge lifts and edge contractions

  • We say that a graph H is a lift-minor of a graph G if H can be obtained from G by a sequence of vertex and edge deletions, edge lifts and contractions

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Summary

Introduction

All graphs in this paper are undirected, loopless, and without multiple edges (unless mentioned otherwise). We say that a graph H is a lift-minor of a graph G if H can be obtained from G by a sequence of vertex and edge deletions, edge lifts and contractions. We identify three conditions on a connected graph G that force any n-vertex graph as a lift-contraction of G. There exists a constant c such that every connected graph G of treewidth at least c · n4 contains every n-vertex graph as a lift-contraction. There exists a function f : N → N such that every connected graph with at least f (n) vertices and minimum degree at least 3 contains every n-vertex graph as a lift-contraction.

Preliminary results
On the structure of lift-contraction-free graphs

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