Related papers: A characterization of $K_{2,4}$-minor-free graphs

Dujmovi\'c et al. [\emph{J.~ACM}~'20] recently proved that every planar graph is isomorphic to a subgraph of the strong product of a bounded treewidth graph and a path. Analogous results were obtained for graphs of bounded Euler genus or…

We introduce the family of $k$-gap-planar graphs for $k \geq 0$, i.e., graphs that have a drawing in which each crossing is assigned to one of the two involved edges and each edge is assigned at most $k$ of its crossings. This definition is…

A graph drawn on the plane is called $1$-plane if each edge is crossed at most once by another edge. In this paper, we show that every $4$-connected $1$-plane graph has a connected spanning plane subgraph. We also show that there exist…

In \emph{smooth orthogonal layouts} of planar graphs, every edge is an alternating sequence of axis-aligned segments and circular arcs with common axis-aligned tangents. In this paper, we study the problem of finding smooth orthogonal…

We show that many graphs with bounded treewidth can be described as subgraphs of the strong product of a graph with smaller treewidth and a bounded-size complete graph. To this end, define the "underlying treewidth" of a graph class…

A graph $G$ is nonseparating projective planar if $G$ has a projective planar embedding without a nonsplit link. Nonseparating projective planar graphs are closed under taking minors and are a superclass of projective outerplanar graphs. We…

For each integer $n\geq 3$, let $W_n$ denote the wheel graph obtained by connecting a single vertex to all vertices of a cycle of length $n$. In particular, $W_6$ is obtained from the Petersen graph by contracting three edges incident with…

Let $G$ be a nontrivial connected graph of order $n$ and $k$ an integer with $2\leq k\leq n$. For a set $S$ of $k$ vertices of $G$, let $\kappa (S)$ denote the maximum number $\ell$ of edge-disjoint trees $T_1,T_2,...,T_\ell$ in $G$ such…

We present an algorithm which can generate all pairwise non-isomorphic $K_2$-hypohamiltonian graphs, i.e. non-hamiltonian graphs in which the removal of any pair of adjacent vertices yields a hamiltonian graph, of a given order. We…

The spread of a graph $G$ is the difference between the largest and smallest eigenvalues of the adjacency matrix of $G$. In this paper, we consider the family of graphs which contain no $K_{2,t}$-minor. We show that for any $t\geq 2$, there…

A k-outerplanar graph is a graph that can be drawn in the plane without crossing such that after k-fold removal of the vertices on the outer-face there are no vertices left. In this paper, we study how to triangulate a k-outerplanar graph…

A graph $G$ is $k$-degenerate if it can be transformed into an empty graph by subsequent removals of vertices of degree $k$ or less. We prove that every connected planar graph with average degree $d \ge 2$ has a 4-degenerate induced…

For planar graphs, it is well known that high connectivity implies a Hamiltonian cycle and hence any 4-connected planar graph has a near-perfect matching. Nevertheless, whether 6-connected 1-planar graphs admit near-perfect matchings…

A graph $G$ is $\{F_{1}, F_{2},\dots,F_{k}\}$-free if $G$ contains no induced subgraph isomorphic to any $F_{i}$ $(1\leq i \leq k)$. A connected graph $G$ is a split graph if its vertex set can be partitioned into a clique and an…

For distinct vertices $u,v$ in a graph $G$, let $\kappa_G(u,v)$ denote the maximum number of internally disjoint $u$-$v$ paths in $G$. Then, $\kappa_G(u,v) \leq \min\{ \mbox{deg}_G(u), \mbox{deg}_G(v) \}$. If equality is attained for every…

It was proved in [Y.-Q. Feng, C. H. Li and J.-X. Zhou, Symmetric cubic graphs with solvable automorphism groups, {\em European J. Combin.} {\bf 45} (2015), 1-11] that a cubic symmetric graph with a solvable automorphism group is either a…

Let $G$ be a graph of order $n$ and let $u,v$ be vertices of $G$. Let $\kappa_G(u,v)$ denote the maximum number of internally disjoint $u$-$v$ paths in $G$. Then the average connectivity $\overline{\kappa}(G)$ of $G$, is defined as $…

We give a characterization of the internally 4-connected binary matroids that have no minor isomorphic to M(K3,3). Any such matroid is either cographic, or is isomorphic to a particular single-element extension of the bond matroid of a…

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…

A graph $G$ is said to be Ramsey size-linear if $r(G,H) =O_G (e(H))$ for every graph $H$ with no isolated vertices. Erd\H{o}s, Faudree, Rousseau, and Schelp observed that $K_4$ is not Ramsey size-linear, but each of its proper subgraphs is,…