Related papers: Linkages in Polytope Graphs

A graph on $2k$ vertices is path-pairable if for any pairing of the vertices the pairs can be joined by edge-disjoint paths. The so far known families of path-pairable graphs have diameter of length at most 3. In this paper we present an…

We identify a family of $O(|E(G)|^2)$ nontrivial facets of the connected matching polytope of a graph $G$, that is, the convex hull of incidence vectors of matchings in $G$ whose covered vertices induce a connected subgraph. Accompanying…

We say that a set of pairs of disjoint cycles $\Lambda(G)$ of a graph $G$ is linked if for any spatial embedding $f$ of $G$ there exists an element $\lambda$ of $\Lambda(G)$ such that the $2$-component link $f(\lambda)$ is nonsplittable,…

The $k$-independence number of a graph is the maximum size of a set of vertices at pairwise distance greater than $k$. A graph is called $k$-partially walk-regular if the number of closed walks of a given length $l\le k$, rooted at a vertex…

A graph is K_{2,3}-saturated if it has no subgraph isomorphic to K_{2,3}, but does contain a K_{2,3} after the addition of any new edge. We prove that the minimum number of edges in a K_{2,3}-saturated graph on n >= 5 vertices is sat(n,…

Let $k,p,q$ be three positive integers. A graph $G$ with order $n$ is said to be $k$-placeable if there are $k$ edge disjoint copies of $G$ in the complete graph on $n$ vertices. A $(p,\,q)$-graph is a graph of order $p$ with $q$ edges.…

In this article we investigate the structure of uniformly $k$-connected and uniformly $k$-edge-connected graphs. Whereas both types have previously been studied independent of each other, we analyze relations between these two classes. We…

In a graph whose vertices are assigned integer ranks, a path is well-ranked if the endpoints have distinct ranks or some interior point has a higher rank than the endpoints. A ranking is an assignment of ranks such that all nontrivial paths…

A graph $G$ is geodetic if between any two vertices there exists a unique shortest path. In 1962 Ore raised the challenge to characterize geodetic graphs, but despite many attempts, such characterization still seems well beyond reach. We…

The complexity of the maximum common connected subgraph problem in partial $k$-trees is still not fully understood. Polynomial-time solutions are known for degree-bounded outerplanar graphs, a subclass of the partial $2$-trees. On the other…

It is known that the vertex connectivity of a planar graph can be computed in linear time. We extend this result to the class of locally maximal 1-plane graphs: graphs that have an embedding with at most one crossing per edge such that the…

Consider a setting where possibly sensitive information sent over a path in a network is visible to every {neighbor} of the path, i.e., every neighbor of some node on the path, thus including the nodes on the path itself. The exposure of a…

Let $D=(V,A)$ be a digraph of order $n$, $S$ a subset of $V$ of size $k$ and $2\le k\leq n$. Strong subgraphs $D_1, \dots , D_p$ containing $S$ are said to be internally disjoint if $V(D_i)\cap V(D_j)=S$ and $A(D_i)\cap A(D_j)=\emptyset$…

A simple topological graph $G$ is a graph drawn in the plane so that any pair of edges have at most one point in common, which is either an endpoint or a proper crossing. $G$ is called saturated if no further edge can be added without…

In this paper we consider the following problem: Over the class of all simple connected graphs of order $n$ with $k$ pendant vertices ($n,k$ being fixed), which graph maximizes (respectively, minimizes) the algebraic connectivity? We also…

A linkage of order k of a graph G is a subgraph with k components, each of which is a path. A linkage is vital if it spans all vertices, and no other linkage connects the same pairs of end vertices. We give a characterization of the graphs…

A graph is a ``$k$-Kuratowski graph'' if it has exactly $k$ components, each isomorphic to $K_5$ or to $K_{3,3}$. We prove that if a graph $G$ contains no $k$-Kuratowski graph as a minor,then there is a set $X$ of boundedly many vertices…

A geometric graph is a graph drawn in the plane so that its vertices and edges are represented by points in general position and straight line segments, respectively. A vertex of a geometric graph is called pointed if it lies outside of the…

The {\em disjointness graph} of a set system is a graph whose vertices are the sets, two being connected by an edge if and only if they are disjoint. It is known that the disjointness graph $G$ of any system of segments in the plane is {\em…

In 2012, Mader conjectured that for any tree $T$ of order $m$, every $k$-connected graph $G$ with minimum degree at least $\lfloor \frac{3k}{2}\rfloor+m-1$ contains a subtree $T'\cong T$ such that $G-V(T')$ remains $k$-connected. In 2022,…