Applications of Order Trees in Infinite Graphs

Infinite Games on Finitely Coloured Graphs with Applications to Automata on Infinite Trees

We examine a class of infinite two-person games on finitely coloured graphs. The main aim is to construct finite memory winning strategies for both players. This problem is motivated by applications to finite automata on infinite trees. A special attention is given to the exact amount of memory needed by the players for their winning strategies. Based on a previous work of Gurevich and Harringt...

On spanning trees and k-connectedness in infinite graphs

If two rays P,Q ⊂ G satisfy (i)–(iii), we call them end-equivalent in G. An end of G is an equivalence class under this relation, and E(G) denotes the set of ends of G. For example, the 2-way infinite ladder has two ends, the infinite grid Z×Z and every infinite complete graph have one end, and the dyadic tree has 20 ends. This paper is concerned with the relationship between the ends of a conn...

On End-faithful Spanning Trees in Infinite Graphs

Let G be an infinite connected graph. A ray (from v) in G is a 1-way infinite path in G (with initial vertex v). An infinite connected subgraph of a ray R ⊂ G is called a tail of R. If X ⊂ G is finite, the infinite component of R\X will be called the tail of R in G\X. The following assertions are equivalent for rays P,Q ⊂ G: (i) There exists a ray R ⊂ G which meets each of P and Q infinitely of...

Topological paths, cycles and spanning trees in infinite graphs

We study topological versions of paths, cycles and spanning trees in infinite graphs with ends that allow more comprehensive generalizations of finite results than their standard notions. For some graphs it turns out that best results are obtained not for the standard space consisting of the graph and all its ends, but for one where only its topological ends are added as new points, while rays ...

Decidability of Second-order Theories and Automata on Infinite Trees