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Matching Complexes of Outerplanar Graphs

Margaret Bayer, Marija Jelić Milutinović, Julianne Vega

Abstract

An outerplanar graph is a planar graph that has a planar drawing with all vertices on the unbounded face. The matching complex of a graph is the simplicial complex whose faces are subsets of disjoint edges of the graph. In this paper we prove that the matching complexes of outerplanar graphs are contractible or homotopy equivalent to a wedge of spheres. This extends known results about trees and polygonal line tilings.

Matching Complexes of Outerplanar Graphs

Abstract

An outerplanar graph is a planar graph that has a planar drawing with all vertices on the unbounded face. The matching complex of a graph is the simplicial complex whose faces are subsets of disjoint edges of the graph. In this paper we prove that the matching complexes of outerplanar graphs are contractible or homotopy equivalent to a wedge of spheres. This extends known results about trees and polygonal line tilings.

Paper Structure

This paper contains 8 theorems, 2 equations, 1 figure.

Key Result

Theorem 2

If $G$ is an outerplanar graph, then the matching complex $\mathcal{M}(G)$ is contractible or homotopy equivalent to a wedge of spheres.

Figures (1)

  • Figure 1: Cases used in proof of Theorem 2

Theorems & Definitions (14)

  • Definition 1
  • Theorem 2
  • Definition 3
  • Proposition 4
  • Proposition 5: Adamaszek adamaszek-split, Theorem 3.3
  • Proposition 6: Engström engstrom2, Lemma 2.4
  • Proposition 7: Engström engstrom-KM, Lemma 2.2
  • Proposition 8: BJV, Proposition 11
  • Definition 9
  • Proposition 10: Harary Harary, pages 106--107
  • ...and 4 more