Related papers: Rooted $K_4$-Minors

We provide a complete structural characterization of $K_{2,4}$-minor-free graphs. The $3$-connected $K_{2,4}$-minor-free graphs consist of nine small graphs on at most eight vertices, together with a family of planar graphs that contains…

In this paper we give structural characterizations of graphs not containing rooted $K_{4}$, $W_{4}$, $K_{2,4}$, and a graph we call $L$.

Let $G$ be a graph and $x_1, x_2, \ldots, x_k$ be distinct vertices of $G$. We say $(G,x_1x_2\ldots x_k)$ has a $C_k$-minor or $G$ has a $C_k$-minor rooted at $x_1x_2\ldots x_k$, if there exist pairwise disjoint sets $X_1, X_2, \ldots,…

A graph G is weakly 4-connected if it is 3-connected, has at least five vertices, and for every pair of sets (A,B) with union V(G) and intersection of size three such that no edge has one end in A-B and the other in B-A, one of the induced…

We first prove that for every vertex x of a 4-connected graph G there exists a subgraph H in G isomorphic to a subdivision of the complete graph K4 on four vertices such that G-V(H) is connected and contains x. This implies an affirmative…

Let ${\rm ex \,} {\mathcal B}$ be a minor-closed class of graphs with a set ${\mathcal B}$ of minimal excluded minors. We study (a) the asymptotic number of graphs without $k+1$ disjoint minors in ${\mathcal B}$ and (b) the properties of a…

Given graphs G and H with V(G) containing V(H), suppose that we have a u,v-path P_{uv} in G for each edge uv in H. There are obvious additional conditions that ensure that G contains H as a rooted subgraph, subdivision, or immersion; we…

We prove that every connected graph $G$ with $m$ edges contains a set $X$ of at most $\frac{3}{16}(m + 1)$ vertices such that $G-X$ has no $K_4$ minor, or equivalently, has treewidth at most $2$. This bound is best possible. Connectivity is…

A complete structural characterization of graphs with no $K_{3,4}$ minor is obtained, and the following consequences are established. Every $4$-connected non-planar graph with at least seven vertices and minimum degree at least five…

Among graphs with 13 edges, there are exactly three internally 4-connected graphs which are $Oct^{+}$, cube+e and $ K_{3,3} +v$. A complete characterization of all 4-connected graphs with no $Oct^{+}$-minor is given in [John Maharry, An…

In this paper we prove two main results about obstruction to graph planarity. One is that, if $G$ is a 3-connected graph with a $K_5$-minor and $T$ is a triangle of $G$, then $G$ has a $K_5$-minor $H$, such that $E(T)\cont E(H)$. Other is…

A fundamental theorem in graph theory states that any 3-connected graph contains a subdivision of $K_4$. As a generalization, we ask for the minimum number of $K_4$-subdivisions that are contained in every $3$-connected graph on $n$…

Results on the existence of various types of spanning subgraphs of graphs are milestones in structural graph theory and have been diversified in several directions. In the present paper, we consider "local" versions of such statements. In…

The present paper is the first one in the sequence of papers about a simple class of {\em framed $4$-graphs}; the goal of the present paper is to collect some well-known results on planarity and to reformulate them in the language of {\em…

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…

Forbidden minors and subdivisions for toroidal graphs are numerous. We consider the toroidal graphs with no $K_{3,3}$-subdivisions that coincide with the toroidal graphs with no $K_{3,3}$-minors. These graphs admit a unique decomposition…

A parallel minor is obtained from a graph by any sequence of edge contractions and parallel edge deletions. We prove that, for any positive integer k, every internally 4-connected graph of sufficiently high order contains a parallel minor…

A graph $G$ is a non-separating planar graph if there is a drawing $D$ of $G$ on the plane such that (1) no two edges cross each other in $D$ and (2) for any cycle $C$ in $D$, any two vertices not in $C$ are on the same side of $C$ in $D$.…

Let $\mathcal{G}$ be a minor-closed graph class. We say that a graph $G$ is a $k$-apex of $\mathcal{G}$ if $G$ contains a set $S$ of at most $k$ vertices such that $G\setminus S$ belongs to $\mathcal{G}.$ We denote by $\mathcal{A}_k…

Let $\mathcal{D}_k$ be the class of graphs for which every minor has minimum degree at most $k$. Then $\mathcal{D}_k$ is closed under taking minors. By the Robertson-Seymour graph minor theorem, $\mathcal{D}_k$ is characterised by a finite…