Related papers: Obstructions of Connectivity 2 for Embedding Graph…
Those maps of a closed surface to the three-dimensional torus that are homotopic to embeddings are characterized. Particular attention is paid to the somewhat intricate case when the surface is nonorientable.
This paper considers *-graphs in which all vertices have degree 4 or 6, and studies the question of calculating the genus of nonorientable surfaces into which such graphs may be embedded. In a previous paper by the authors, the problem of…
Unlike minors, the induced subgraph obstructions to bounded treewidth come in a large variety, including, for every $t\geq 1$, the $t$-basic obstructions: the graphs $K_{t+1}$ and $K_{t,t}$, along with the subdivisions of the $t$-by-$t$…
Nancy G. Kinnersley and Michael A. Langston has determined the excluded minors for the class of graphs with path-width at most two by computer. Their list consisted of 110 graphs. Such a long list is difficult to handle and gives no insight…
In [36, Section 8], the present author proposed the hypergraph obstruction for the existence of k-regular embeddings. In this paper, we develop the hypergraph obstruction concretely and give some homological obstructions for the k-regular…
The notion of forbidden-transition graphs allows for a robust generalization of walks in graphs. In a forbidden-transition graph, every pair of edges incident to a common vertex is permitted or forbidden; a walk is compatible if all pairs…
We present a simple combinatorial model for quasipositive surfaces and positive braids, based on embedded bipartite graphs. As a first application, we extend the well-known duality on standard diagrams of torus links to twisted torus links.…
Visual rendering of graphs is a key task in the mapping of complex network data. Although most graph drawing algorithms emphasize aesthetic appeal, certain applications such as travel-time maps place more importance on visualization of…
The coloring problem is studied in the paper for graph classes defined by two small forbidden induced subgraphs. We prove some sufficient conditions for effective solvability of the problem in such classes. As their corollary we determine…
If a graph can be drawn on the torus so that every two independent edges cross an even number of times, then the graph can be embedded on the torus.
Testing a graph on 2-vertex- and 2-edge-connectivity are two fundamental algorithmic graph problems. For both problems, different linear-time algorithms with simple implementations are known. Here, an even simpler linear-time algorithm is…
A graph is sub-unicyclic if it contains at most one cycle. We also say that a graph $G$ is $k$-apex sub-unicyclic if it can become sub-unicyclic by removing $k$ of its vertices. We identify 29 graphs that are the minor-obstructions of the…
We address the problem of necessary conditions and topological obstructions for the existence of robustly transitive maps on surfaces. Concretely, we show that partial hyperbolicity is a necessary condition in order to have $C^1$ robustly…
We consider closed and orientable immersed hypersurfaces of translational manifolds. Given a vector field on such a hypersurface, we define a perturbation of its Gauss map, which allows us to obtain topological invariants for the immersion…
In this paper we characterize the unique graph whose algebraic connectivity is minimum among all connected graphs with given order and fixed matching number or edge covering number, and present two lower bounds for the algebraic…
An obstacle representation of a graph $G$ is a set of points in the plane representing the vertices of $G$, together with a set of polygonal obstacles such that two vertices of $G$ are connected by an edge in $G$ if and only if the line…
The sphere $S^2$ and the torus $T^2$ are the only closed connected surfaces for which higher topological complexities are known (for each $n\in\{2,3,...\}\subset\mathbb{N}$, $\mathrm{TC}_n(S^2)=n$ and $\mathrm{TC}_n(T^2)=2n-2$). This text…
The connectivity of a graph is an important parameter to evaluate its reliability. $k$-restricted connectivity (resp. $R^h$-restricted connectivity) of a graph $G$ is the minimum cardinality of a set $S$ of vertices in $G$, if exists, whose…
We showed in another paper [arXiv:1103.1759] that every connected graph can be realized as the cut locus of some point on some riemannian surface $S$. Here, criteria for the orientability of $S$ are given, and are applied to classify the…
A forbidden transition graph is a graph defined together with a set of permitted transitions i.e. unordered pair of adjacent edges that one may use consecutively in a walk in the graph. In this paper, we look for the smallest set of…