End (graph theory)

Ends of graphs were defined by Rudolf Halin (1964) in terms of equivalence classes of infinite paths.[1] A ray in an infinite graph is a semi-infinite simple path; that is, it is an infinite sequence of vertices ${\displaystyle v_{0},v_{1},v_{2},\dots }$ in which each vertex appears at most once in the sequence and each two consecutive vertices in the sequence are the two endpoints of an edge in the graph. According to Halin's definition, two rays ${\displaystyle r_{0}}$ and ${\displaystyle r_{1}}$ are equivalent if there is a ray ${\displaystyle r_{2}}$ (which may equal one of the two given rays) that contains infinitely many of the vertices in each of ${\displaystyle r_{0}}$ and ${\displaystyle r_{1}}$. This is an equivalence relation: each ray is equivalent to itself, the definition is symmetric with regard to the ordering of the two rays, and it can be shown to be transitive. Therefore, it partitions the set of all rays into equivalence classes, and Halin defined an end as one of these equivalence classes.[2]

An alternative definition of the same equivalence relation has also been used: two rays ${\displaystyle r_{0}}$ and ${\displaystyle r_{1}}$ are equivalent if there is no finite set ${\displaystyle X}$ of vertices that separates infinitely many vertices of ${\displaystyle r_{0}}$ from infinitely many vertices of ${\displaystyle r_{1}}$.[3] This is equivalent to Halin's definition: if the ray ${\displaystyle r_{2}}$ from Halin's definition exists, then any separator must contain infinitely many points of ${\displaystyle r_{2}}$ and therefore cannot be finite, and conversely if ${\displaystyle r_{2}}$ does not exist then a path that alternates as many times as possible between ${\displaystyle r_{0}}$ and ${\displaystyle r_{1}}$ must form the desired finite separator.

Ends also have a more concrete characterization in terms of havens, functions that describe evasion strategies for pursuit–evasion games on a graph ${\displaystyle G}$.[4] In the game in question, a robber is trying to evade a set of policemen by moving from vertex to vertex along the edges of ${\displaystyle G}$. The police have helicopters and therefore do not need to follow the edges; however the robber can see the police coming and can choose where to move next before the helicopters land. A haven is a function ${\displaystyle \beta }$ that maps each set ${\displaystyle X}$ of police locations to one of the connected components of the subgraph formed by deleting ${\displaystyle X}$; a robber can evade the police by moving in each round of the game to a vertex within this component. Havens must satisfy a consistency property (corresponding to the requirement that the robber cannot move through vertices on which police have already landed): if ${\displaystyle X}$ is a subset of ${\displaystyle Y}$, and both ${\displaystyle X}$ and ${\displaystyle Y}$ are valid sets of locations for the given set of police, then ${\displaystyle \beta (X)}$ must be a superset of ${\displaystyle \beta (Y)}$. A haven has order ${\displaystyle k}$ if the collection of police locations for which it provides an escape strategy includes all subsets of fewer than ${\displaystyle k}$ vertices in the graph; in particular, it has order ${\displaystyle \aleph _{0}}$ (the smallest aleph number) if it maps every finite subset ${\displaystyle X}$ of vertices to a component of ${\displaystyle G\setminus X}$. Every ray in ${\displaystyle G}$ corresponds to a haven of order ${\displaystyle \aleph _{0}}$, namely, the function ${\displaystyle \beta }$; that maps every finite set ${\displaystyle X}$ to the unique component of ${\displaystyle G\setminus X}$ that contains infinitely many vertices of the ray. Conversely, every haven of order ${\displaystyle \aleph _{0}}$ can be defined in this way by a ray.[5] Two rays are equivalent if and only if they define the same haven, so the ends of a graph are in one-to-one correspondence with its havens of order ${\displaystyle \aleph _{0}}$.[4]

Part of an infinite grid graph, with vertices at the points where two grid lines meet. Despite having many different rays, it has only one end.

If the infinite graph ${\displaystyle G}$ is itself a ray, then it has infinitely many ray subgraphs, one starting from each vertex of ${\displaystyle G}$. However, all of these rays are equivalent to each other, so ${\displaystyle G}$ only has one end.

If ${\displaystyle G}$ is a forest (that is, a graph with no finite cycles), then the intersection of any two rays is either a path or a ray; two rays are equivalent if their intersection is a ray. If a base vertex is chosen in each connected component of ${\displaystyle G}$, then each end of ${\displaystyle G}$ contains a unique ray starting from one of the base vertices, so the ends may be placed in one-to-one correspondence with these canonical rays. Every countable graph ${\displaystyle G}$ has a spanning forest with the same set of ends as ${\displaystyle G}$.[6] However, there exist uncountably infinite graphs with only one end in which every spanning tree has infinitely many ends.[7]

If ${\displaystyle G}$ is an infinite grid graph, then it has many rays, and arbitrarily large sets of vertex-disjoint rays. However, it has only one end. This may be seen most easily using the characterization of ends in terms of havens: the removal of any finite set of vertices leaves exactly one infinite connected component, so there is only one haven (the one that maps each finite set to the unique infinite connected component).

In point-set topology, there is a concept of an end that is similar to, but not quite the same as, the concept of an end in graph theory, dating back much earlier to Freudenthal (1931). If a topological space can be covered by a nested sequence of compact sets ${\displaystyle \kappa _{0}\subset \kappa _{1}\subset \kappa _{2}\dots }$, then an end of the space is a sequence of components ${\displaystyle U_{0}\supset U_{1}\supset U_{2}\dots }$ of the complements of the compact sets. This definition does not depend on the choice of the compact sets: the ends defined by one such choice may be placed in one-to-one correspondence with the ends defined by any other choice.

An infinite graph ${\displaystyle G}$ may be made into a topological space in two different but related ways:

• Replacing each vertex of the graph by a point and each edge of the graph by an open unit interval produces a Hausdorff space from the graph in which a set ${\displaystyle S}$ is defined to be open whenever each intersection of ${\displaystyle S}$ with an edge of the graph is an open subset of the unit interval.
• Replacing each vertex of the graph by a point and each edge of the graph by a point produces a non-Hausdorff space in which the open sets are the sets ${\displaystyle S}$ with the property that, if a vertex ${\displaystyle v}$ of ${\displaystyle G}$ belongs to ${\displaystyle S}$, then so does every edge having ${\displaystyle v}$ as one of its endpoints.

In either case, every finite subgraph of ${\displaystyle G}$ corresponds to a compact subspace of the topological space, and every compact subspace corresponds to a finite subgraph together with, in the Hausdorff case, finitely many compact proper subsets of edges. Thus, a graph may be covered by a nested sequence of compact sets if and only if it is locally finite, having a finite number of edges at every vertex.

If a graph ${\displaystyle G}$ is connected and locally finite, then it has a compact cover in which the set ${\displaystyle \kappa _{i}}$ is the set of vertices at distance at most ${\displaystyle i}$ from some arbitrarily chosen starting vertex. In this case any haven ${\displaystyle \beta }$ defines an end of the topological space in which ${\displaystyle U_{i}=\beta (\kappa _{i})}$. And conversely, if ${\displaystyle U_{0}\supset U_{1}\supset U_{2}\dots }$ is an end of the topological space defined from ${\displaystyle G}$, it defines a haven in which ${\displaystyle \beta (X)}$ is the component containing ${\displaystyle U_{i}}$, where ${\displaystyle i}$ is any number large enough that ${\displaystyle \kappa _{i}}$ contains ${\displaystyle X}$. Thus, for connected and locally finite graphs, the topological ends are in one-to-one correspondence with the graph-theoretic ends.[8]

For graphs that may not be locally finite, it is still possible to define a topological space from the graph and its ends. This space can be represented as a metric space if and only if the graph has a normal spanning tree, a rooted spanning tree such that each graph edge connects an ancestor-descendant pair. If a normal spanning tree exists, it has the same set of ends as the given graph: each end of the graph must contain exactly one infinite path in the tree.[9]