Graphs Without Dead Ends

Traps, Invariants, and Dead-Ends

We consider the problem of deriving formulas that capture traps, invariants, and dead-ends in classical planning through polynomial forms of preprocessing. An invariant is a formula that is true in the initial state and in all reachable states. A trap is a conditional invariant: once a state is reached that makes the trap true, all the states that are reachable from it will satisfy the trap for...

African Development: Dead Ends and New Beginnings

Frobenius Problem and Dead Ends in Integers

Let a and b be positive, relatively prime integers. We show that the following are equivalent: (i) d is a dead end in the (symmetric) Cayley graph of Z with respect to a and b, (ii) d is a Frobenius value with respect to a and b (it cannot be written as a non-negative or non-positive integer linear combination of a and b), and d is maximal (in the Cayley graph) with respect to this property. In...

Introduction to Ends of Graphs

We give an elementary introduction to the theory of ends of locally finite graphs for students and mathematicians from other fields. The focus is on basic topological properties of the end space. We will also discuss Cayley graphs and some connections between ends and group actions on graphs.

Edge-Ends in Countable Graphs

The notion of ends equivalence classes on the set of rays (one-way infinite paths) of a graph is one of the most studied topics in infinite graph theory. An introduction to this theory and basic results can be found in Halin [3]. Halin defined two rays to be equivalent if no finite set of vertices can separate an infinite part of the first ray from an infinite part of the second one. In particu...