Related papers: Links in projective planar graphs

A graph is intrinsically projectively linked (IPL) if its every embedding in projective space contains a nonsplit link. Some minor-minimal IPL graphs have been found previously. We determine that no minor-minimal IPL graphs on 16 edges…

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 $D$.…

We exhibit several families of planar graphs that are minor-minimal intrinsically spherical $3$-linked. A graph is intrinsically spherical 3-linked if it is planar graph that has, in every spherical embedding, a non-split 3-link consisting…

A planar graph $G$ is said to be non-separating if there exists an embedding of $G$ in $\mathbb{R}^2$ such that for any cycle $\mathcal{C}\subset G$, all vertices of $G\setminus \mathcal{C}$ are within the same connected component of…

We construct a graph G such that any embedding of G into R^{3} contains a nonsplit link of two components, where at least one of the components is a nontrivial knot. Further, for any m < n we produce a graph H so that every embedding of H…

It is shown that a simple graph which is embeddable in the real projective plane is minimally 3-rigid if and only if it is (3,6)-tight. Moreover the topologically uncontractible embedded graphs of this type are constructible from one of 8…

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…

Let $G$ be a group. The intersection graph of subgroups of $G$, denoted by $\mathscr{I}(G)$, is a graph with all the proper subgroups of $G$ as its vertices and two distinct vertices in $\mathscr{I}(G)$ are adjacent if and only if the…

We examine graphs that contain a non-trivial link in every embedding into real projective space, using a weaker notion of unlink than was used by Flapan, et al. We call such graphs intrinsically linked in projective space. We fully…

A graph G is intrinsically S^1-linked if for every embedding of the vertices of G into S^1, vertices that form the endpoints of two disjoint edges in G form a non-split link in the embedding. We show that a graph is intrinsically S^1-linked…

We say that a graph is intrinsically non-trivial if every spatial embedding of the graph contains a non-trivial spatial subgraph. We prove that an intrinsically non-trivial graph is intrinsically linked, namely every spatial embedding of…

A pair of non-adjacent edges is said to be separated in a circular ordering of vertices, if the endpoints of the two edges do not alternate in the ordering. The circular separation dimension of a graph $G$, denoted by $\pi^\circ(G)$, is the…

A nonplanar graph G is called almost-planar if for every edge e of G, at least one of G\e and G/e is planar. In 1990, Gubser characterized 3-connected almost-planar graphs in his dissertation. However, his proof is so long that only a small…

Median graphs are connected graphs in which for all three vertices there is a unique vertex that belongs to shortest paths between each pair of these three vertices. In this paper we provide several novel characterizations of planar median…

We say that a graph $G$ is $(2,m)$-linked if, for any distinct vertices $a_1,\ldots, a_m, b_1,b_2$ in $G$, there exist vertex disjoint connected subgraphs $A,B$ of $G$ such that $\{a_1, \ldots, a_m\}$ is contained in $A$ and $\{b_1,b_2\}$…

We say that a graph is intrinsically knotted or completely 3-linked if every embedding of the graph into the 3-sphere contains a nontrivial knot or a 3-component link any of whose 2-component sublink is nonsplittable. We show that a graph…

A subset of vertices of a graph is minimal if, within all subsets of the same size, its vertex boundary is minimal. We give a complete, geometric characterization of minimal sets for the planar integer lattice X. Our characterization…

A partially embedded graph (or PEG) is a triple (G,H,\H), where G is a graph, H is a subgraph of G, and \H is a planar embedding of H. We say that a PEG (G,H,\H) is planar if the graph G has a planar embedding that extends the embedding \H.…

Kuratowski's theorem says that the minimal (under subgraph containment) graphs that are not planar are the subdivisions of $K_5$ and of $K_{3,3}$. Here we study the minimal (under subdigraph containment) strongly-connected digraphs that are…

A graph is 1-planar if it can be drawn on the plane so that each edge is crossed by no more than one other edge (and any pair of crossing edges cross only once). A non-1-planar graph $G$ is minimal if the graph $G-e$ is 1-planar for every…