Related papers: Graph product structure for non-minor-closed class…
In this paper we show that every graph of pathwidth less than $k$ that has a path of order $n$ also has an induced path of order at least $\frac{1}{3} n^{1/k}$. This is an exponential improvement and a generalization of the polylogarithmic…
We prove that for every countable string graph $S$, there is a planar graph $G$ with $V(G)=V(S)$ such that \[ \frac{1}{23660800}d_S(u,v) \le d_G(u,v) \le 162 d_S(u,v) \] for all $u,v\in V(S)$, where $d_S(u,v)$, $d_G(u,v)$ denotes the…
We consider relations between the size, treewidth, and local crossing number (maximum number of crossings per edge) of graphs embedded on topological surfaces. We show that an $n$-vertex graph embedded on a surface of genus $g$ with at most…
A graph $A$ is "apex" if $A-z$ is planar for some vertex $z\in V(A)$. Eppstein [Algorithmica, 2000] showed that for a minor-closed class $\mathcal{G}$, the graphs in $\mathcal{G}$ with bounded radius have bounded treewidth if and only if…
We call a finite undirected graph minimally k-matchable if it has at least k distinct perfect matchings but deleting any edge results in a graph which has not. An odd subdivision of some graph G is any graph obtained by replacing every edge…
The treewidth is a structural parameter that measures the tree-likeness of a graph. Many algorithmic and combinatorial results are expressed in terms of the treewidth. In this paper, we study the treewidth of outer $k$-planar graphs, that…
A graph on at least ${{k+1}}$ vertices is uniformly $k$-connected if each pair of its vertices is connected by $k$ and not more than $k$ independent paths. We reinvestigate a recent constructive characterization of uniformly $3$-connected…
A graph class $\mathcal{G}$ has linear growth if, for each graph $G \in \mathcal{G}$ and every positive integer $r$, every subgraph of $G$ with radius at most $r$ contains $O(r)$ vertices. In this paper, we show that every graph class with…
A graph $G$ is $k$-edge geodetic graph if every edge of $G$ lies in at least one geodesic of length $k$. We studied some basic properties of $k$-edge geodetic graphs. We investigated the $k$ edge-geodeticity of complete bipartite graph…
We prove that every sufficiently big 6-connected graph of bounded tree-width either has a K_6 minor, or has a vertex whose deletion makes the graph planar. This is a step toward proving that the same conclusion holds for all sufficiently…
We show that every $H$-minor-free graph that also excludes a $(k \times k)$-grid as a minor has treewidth/branchwidth bounded from above by a function $f(t,k)$ that is linear in $k$ and polynomial in $t := |V(H)|$. Such a result was proven…
A graph is \emph{fan-crossing free} if it has a drawing in the plane so that each edge is crossed by independent edges, that is the crossing edges have distinct vertices. On the other hand, it is \emph{fan-crossing} if the crossing edges…
Half graphs and their variants, such as ladders, semi-ladders and co-matchings, are combinatorial objects that encode total orders in graphs. Works by Adler and Adler (Eur. J. Comb.; 2014) and Fabia\'nski et al. (STACS; 2019) prove that in…
Treewidth is a parameter that emerged from the study of minor closed classes of graphs (i.e. classes closed under vertex and edge deletion, and edge contraction). It in some sense describes the global structure of a graph. Roughly, a graph…
We prove that a connected, locally finite, quasi-transitive graph which is quasi-isometric to a planar graph is necessarily accessible. This leads to a complete classification of the finitely generated groups which are quasi-isometric to…
The Kuratowski-Wagner Theorem asserts that a graph is planar if and only if it does not have either $K_{3,3}$ or $K_5$ as a minor. Using this Wagner obtained a precise description of all graphs with no $K_{3,3}$ minor and all graphs with no…
The $2$-layer drawing model is a well-established paradigm to visualize bipartite graphs. Several beyond-planar graph classes have been studied under this model. Surprisingly, however, the fundamental class of $k$-planar graphs has been…
Classes with bounded expansion, which generalise classes that exclude a topological minor, have recently been introduced by Ne\v{s}et\v{r}il and Ossona de Mendez. These classes are defined by the fact that the maximum average degree of a…
In this paper we investigate the connectedness and the isomorphism problems for zig-zag products of two graphs. A sufficient condition for the zig-zag product of two graphs to be connected is provided, reducing to the study of the…
There has been much recent interest in random graphs sampled uniformly from the n-vertex graphs in a suitable minor-closed class, such as the class of all planar graphs. Here we use combinatorial and probabilistic methods to investigate a…