Graphs without isometric rays and invariant subgraph properties, I
✍ Scribed by Polat, Norbert
- Publisher
- John Wiley and Sons
- Year
- 1998
- Tongue
- English
- Weight
- 190 KB
- Volume
- 27
- Category
- Article
- ISSN
- 0364-9024
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✦ Synopsis
there exists an element a of A -{x} such that the interval (set of vertices of all shortest paths) between x and a is disjoint from S. A set A ⊆ V (G) is geodesically closed if it contains all vertices which geodesically dominate A. These geodesically closed sets define a topology, called the geodesic topology, on V (G). We prove that a connected graph G has no isometric rays if and only if the set V (G) endowed with the geodesic topology is compact, or equivalently if and only if the vertex set of every ray in G is geodesically dominated.
We prove different invariant subgraph properties for graphs containing no isometric rays. In particular we show that every self-contraction (map which preserves or contracts the edges) of a chordal graph G stabilizes a non-empty finite simplex (complete graph) if and only if G is connected and contains no isometric rays and no infinite simplices.