The graphs that have Hadwiger number at most four have been characterized by Wagner (1937). The graphs with any finite bound on the Hadwiger number are sparse, and have small chromatic number. Determining the Hadwiger number of a graph is NP-hard but fixed-parameter tractable.

A graph *G* has Hadwiger number at most two if and only if it is a forest, for a three-vertex complete minor can only be formed by contracting a cycle in *G*.

A graph has Hadwiger number at most three if and only if its treewidth is at most two, which is true if and only if each of its biconnected components is a series-parallel graph.

Wagner's theorem, which characterizes the planar graphs by their forbidden minors, implies that the planar graphs have Hadwiger number at most four. In the same paper that proved this theorem, Wagner (1937) also characterized the graphs with Hadwiger number at most four more precisely: they are graphs that can be formed by clique-sum operations that combine planar graphs with the eight-vertex Wagner graph.

The graphs with Hadwiger number at most five include the apex graphs and the linklessly embeddable graphs, both of which have the complete graph *K*_{6} among their forbidden minors.^{}

This page was last edited on 18 February 2016, at 21:33.

Reference: https://en.wikipedia.org/wiki/Hadwiger_number under CC BY-SA license.

Reference: https://en.wikipedia.org/wiki/Hadwiger_number under CC BY-SA license.

- Graph Theory
- Undirected Graph
- Complete Graph
- Contracting Edges
- Minor
- Hugo Hadwiger
- Hadwiger Conjecture
- Chromatic Number
- Wagner (1937)
- NP-hard
- Fixed-parameter Tractable
- Forest
- Cycle
- Treewidth
- Biconnected Components
- Series-parallel Graph
- Wagner's Theorem
- Planar Graphs
- Forbidden Minors
- Wagner (1937)
- Clique-sum
- Wagner Graph
- Apex Graphs
- Linklessly Embeddable Graphs

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