On topological diversity of rectilinear representations of countably infinite graphs

Abstract

We prove that, for each simple graph G whose set of vertices is countably infinite, there is a family $${\varvec{\mathcal{R}}(\varvec{G})}$$ of the cardinality of the continuum of graphs such that (1) each graph $${\varvec{H} \in \varvec{\mathcal{R}}(\varvec{G})}$$ is isomorphic to G, all vertices of H are points of the Euclidean space E 3, all edges of H are straight line segments (the ends of each edge are the vertices joined by it), the intersection of any two edges of H is either their common vertex or empty, and any isolated vertex of H does not belong to any edge of H; (2) all sets $${\varvec{\mathcal{B}}(\varvec{H})}$$ ( $${\varvec{H} \in \varvec{\mathcal{R}}(\varvec{G})}$$ ), where $${\varvec{\mathcal{B}}(\varvec{H})\subset \mathbf{E}^3}$$ is the union of all vertices and all edges of H, are pairwise not homeomorphic; moreover, for any graphs $${\varvec{H}_1 \in \varvec{\mathcal{R}}(\varvec{G})}$$ and $${\varvec{H}_2 \in \varvec{\mathcal{R}}(\varvec{G})}$$ , $${\varvec{H}_1 \ne \varvec{H}_2}$$ , and for any finite subsets $${\varvec{S}_i \subset \varvec{\mathcal{B}}(\varvec{H}_i)}$$ (i = 1, 2), the sets $${\varvec{\mathcal{B}}(\varvec{H}_1){\setminus} \varvec{S}_1}$$ and $${\varvec{\mathcal{B}}(\varvec{H}_2){\setminus} \varvec{S}_2}$$ are not homeomorphic.