Graphs without isometric rays and invariant subgraph properties, I

A ray of a graph G is isometric if every path in R is a shortest path in G. A vertex x of Ggeodesically dominates a subset A of V(G) if, for every finite S ⊆ V(G − x), 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. © 1998 John Wiley & Sons, Inc. J Graph Theory 27: 99–109, 1998