Related papers: On Induced Versions of Menger's Theorem on Sparse …

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 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…

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,…

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 prove that there exist functions $f$ and $g$ such that for all positive integers $k$ and $d$, for every graph $G$ and every subset $A$ of the vertices of $G$, either $G$ contains $k$ $A$-paths such that vertices of different $A$-paths…

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…

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…

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…

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…

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…

We prove Menger-type results in which the obtained paths are pairwise non-adjacent, both for graphs of bounded maximum degree and, more generally, for graphs excluding a topological minor. We further show better bounds in the subcubic case,…

One of the most basic results in graph theory states that every graph with at least two vertices has two vertices with the same degree. Since there are graphs without $3$ vertices of the same degree, it is natural to ask if for any fixed…

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…

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…

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…

The classical Menger's theorem states that in any undirected (or directed) graph $G$, given a pair of vertices $s$ and $t$, the maximum number of vertex (edge) disjoint paths is equal to the minimum number of vertices (edges) needed to…

The classical Erd\H{o}s-P\'{o}sa theorem states that for each positive integer k there is an f(k) such that, in each graph G which does not have k+1 disjoint cycles, there is a blocker of size at most f(k); that is, a set B of at most f(k)…

A $k$-block in a graph $G$ is a maximal set of at least $k$ vertices no two of which can be separated in $G$ by deleting fewer than $k$ vertices. The block number $\beta(G)$ of $G$ is the maximum integer $k$ for which $G$ contains a…

Dirac (1952) proved that every connected graph of order $n>2k+1$ with minimum degree more than $k$ contains a path of length at least $2k+1$. Erd\H{o}s and Gallai (1959) showed that every $n$-vertex graph $G$ with average degree more than…