Related papers: On an induced version of Menger's theorem

We prove that Menger's theorem is valid for infinite graphs, in the following strong form: let $A$ and $B$ be two sets of vertices in a possibly infinite digraph. Then there exist a set $\cp$ of disjoint $A$-$B$ paths, and a set $S$ of…

We give an approximate Menger-type theorem for when a graph $G$ contains two $X-Y$ paths $P_1$ and $P_2$ such that $P_1 \cup P_2$ is an induced subgraph of $G$. More generally, we prove that there exists a function $f(d) \in O(d)$, such…

Menger's theorem - the maximum number of vertex-disjoint $X$-$Y$ paths is equal to the minimum size of an $X$-$Y$ separator - is generally not true in bidirected graphs. We prove that Menger's theorem holds true if we take the nontrivial…

Bidirected graphs are a generalisation of directed graphs that arises in the study of undirected graphs with perfect matchings. Menger's famous theorem - the minimum size of a set separating two vertex sets $X$ and $Y$ is the same as the…

Let $A$ and $B$ be sets of vertices in a graph $G$. Menger's theorem states that for every positive integer $k$, either there exists a collection of $k$ vertex-disjoint paths between $A$ and $B$, or $A$ can be separated from $B$ by a set of…

Menger's Theorem is a fundamental result in graph theory. It states that if in a graph $G$ with distinguished sets of terminal vertices $S$ and $T$ there are no $k$ pairwise vertex-disjoint $S$-$T$ paths, then there is a set of less than…

Menger's Edge Theorem asserts that there exist $k$ pairwise edge-disjoint paths between two vertices in an undirected graph if and only if a deletion of any $k-1$ or less edges does not disconnect these two vertices. Alternatively, there…

Menger's well-known theorem from 1927 characterizes when it is possible to find $k$ vertex-disjoint paths between two sets of vertices in a graph $G$. Recently, Georgakopoulos and Papasoglu and, independently, Albrechtsen, Huynh, Jacobs,…

Menger's theorem says that, for $k\ge0$, if $S, T$ are sets of vertices in a graph $G$, then either there are $k + 1$ vertex-disjoint paths between $S$ and $T$, or there is a set X of at most $k$ vertices such that every $S$-$T$ path passes…

Infinite graphs are finitary in the sense that their points are connected via finite paths. So what would an infinitary generalization of finite graphs look like? Usually this question is answered with the aid of topology, e.g. in the case…

Polat generalised Menger's theorem -- the maximum number of vertex-disjoint paths between two sets $A$ and $B$ equals the minimum size of an $A$-$B$ separator -- to ends of undirected graphs. In this paper we extend Menger's theorem to ends…

Menger's theorem tells us that if $S,T$ are sets of vertices in a graph $G$, then (for $k\ge0$) either there are $k+1$ vertex-disjoint paths between $S$ and $T$, or there is a set of $k$ vertices separating $S$ and $T$. But what if we want…

The Known Menger's theorem states that in a finite graph, the size of a minimum separator set of any pair of vertices is equal to the maximum number of disjoint paths that can be found between these two vertices. In this paper, we study the…

Answering a question of Diestel, we develop a topological notion of gammoids in infinite graphs which, unlike traditional infinite gammoids, always define a matroid. As our main tool, we prove for any infinite graph $G$ with vertex sets $A$…

A tuple (s1,t1,s2,t2) of vertices in a simple undirected graph is 2-linked when there are two vertex-disjoint paths respectively from s1 to t1 and s2 to t2. A graph is 2-linked when all such tuples are 2-linked. We give a new and simple…

We represent the universal Menger curve as the topological realization $|\mathbb{M}|$ of the projective Fra\"iss\'e limit ${\mathbb M}$ of the class of all finite connected graphs. We show that $\mathbb{M}$ satisfies combinatorial analogues…

We prove that a cubic graph with $m$ edges has an induced matching with at least $m/9$ edges. Our result generalizes a result for planar graphs due to Kang, Mnich, and M\"{u}ller (Induced matchings in subcubic planar graphs, SIAM J.…

Coarse graph theory concerns finding 'coarse' analogues of graph theory theorems, replacing disjointness with being far apart. One of the most interesting open questions is to find a coarse analogue of Menger's theorem, which characterizes…

Over all graphs (or unicyclic graphs) of a given order, we characterise those graphs that minimise or maximise the number of connected induced subgraphs. For each of these classes, we find that the graphs that minimise the number of…

Menger's theorem is an important building block of numerous results in the study of graph structure. We consider a variant in terms of coarse geometry. We say that a set of graphs has the weak coarse Menger property if there exist functions…