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3
votes
3answers
102 views

Complexity of isotopy of embedded graphs

I am looking for previous work on the following problem: given two graphs embedded in the plane without crossing, determine if they are isotopic. By isotopic I mean that there is a continuous ...
0
votes
1answer
73 views

Constrained Topological Sorting with bounded number of chains

In general, constrained topological sorting is NP-hard. Now we add another constraint to it, such that take any k+1 nodes and there will be at least one pair ...
-5
votes
1answer
72 views

Does any DAG can be topologically sorted? [closed]

I am not good enough in computer science. My intention is to solve some programming problem in terms of DAG's. The key point is that before getting them into database, I need run "topological sort" in ...
4
votes
1answer
100 views

Properties of toroidal graph

I am interested in work pertaining to graphs that have genus 1 i.e. toroidal graphs. Specifically, i am trying to find answers to the questions below. Since toroidal graphs can be recognized in $P$ , ...
6
votes
0answers
94 views

Structures obtained by gluing simplices

I'm looking for the correct name of geometric structures obtained as follows. 2-structures: A collection $X$ of triangles is a $2$-structure. If $X$ is a $2$-structure and $Y$ is obtained from $X$ ...
20
votes
1answer
293 views

Minor closed properties that are explicitly MSO expressible

Below, MSO denotes the monadic second order logic of graphs with vertex-set and edge-set quantifications. Let $\mathcal{F}$ be a minor closed family of graphs. It follows from Robertson and Seymour'...
14
votes
3answers
411 views

Applications for set theory, ordinal theory, infinite combinatorics and general topology in computer science?

I am a mathematician interested in set theory, ordinal theory, infinite combinatorics and general topology. Are there any applications for these subjects in computer science? I have looked a bit, and ...
1
vote
0answers
117 views

Example of non-disk bounding planarly nested sequences of cycles

I am trying to find an example for the Theorem 5.1 of the paper "Combinatorial Local Planarity and the Width of Graph Embeddings" that can be found at http://www.fmf.uni-lj.si/~mohar/Reprints/1992/...
4
votes
1answer
148 views

Addding edges to spanning tree without destroying planarity

Given a graph $G=(V,E)$ with n vertices, m edges, and the maximum degree $\Delta$. Let $T$ be a spanning tree of $G$. Let $E_c \subseteq E - E(T)$ be the maximum number of edges that we can add to $T$ ...
-1
votes
1answer
866 views

All pairs shortest paths in a DAG [closed]

I have studied the Floyd-Warshall and Johnson algorithms. I am trying to understand if the all pairs shortest paths research in a directed graph G can be implemented in a more efficient way if I ...
3
votes
1answer
308 views

Topological sort with constraints on the relative difference between the vertex labels

A topological sort of a graph $G(V,E)$ consisting of $n$ vertices assigns a label $L(v_x)$ to a vertex $v_x$ where $L$ is defined as $L:V \rightarrow \{1,\dots,n\}$. Let additional constraints over ...
1
vote
1answer
348 views

combinatorical embedding

I have a problem with the following statement : Every combinatorial embedding is equivalent to one with $\lambda(T) = 1$ on a spanning tree of G What does this mean ? OK in a spanning tree there ...
6
votes
1answer
157 views

Equivalent embeddings of a graph

I have difficulties finding a good definition of two embeddings of a (planar) graph in the plane being equivalent. Intuitively I mean by equivalent that the embeddings look the same up to ...
11
votes
2answers
231 views

Approximability of the genus problem

What is currently known about the approximability of the genus problem? A preliminary search tells me that a constant factor approximation is trivial for sufficiently dense graphs, and an $n^\epsilon$-...
3
votes
0answers
39 views

Polyhedral embedding from graph degree sequence

Given: A degree sequence. Wanted: A graph and a polyhedral embedding of this graph (described by a rotation system or something equivalent). By polyhedral embedding I mean only the combinatorial ...
5
votes
1answer
360 views

Face-walks in rotation systems for graphs

Given a graph $G$, a rotation system for $G$ is composed of two elements: $\pi = \{\pi_v: v\in V(G)\}$, where $\pi_v$ is a cyclic permutation of the edges incident on $v$. Thus if $e$ is an edge ...
2
votes
1answer
77 views

Is the dual of a polyhedral embedding a polyhedral embedding?

A polyhedral embedding of a graph on a surface is an embedding without edge crossings such that all the faces are bounded by simple cycles, and any two faces share a common vertex, share a common edge,...
7
votes
1answer
188 views

Algorithm to find a polyhedral embedding

A polyhedral embedding of a graph on a surface is an embedding without edge crossings such that all the faces are bounded by simple cycles, and any two faces share a common vertex, share a common edge,...
5
votes
1answer
197 views

Finding a simple dual of a simple graph in some surface

Given a cellular embedding of a graph on a surface (by 'surface' I mean here a sphere with some $n\geq 0$ handles), one can define a dual multigraph by treating the faces of the original graph ...
10
votes
1answer
650 views

Finding a dual of a graph

According to the book Topological Graph Theory by Gross and Tucker, given a cellular embedding of a graph on a surface (by 'surface' I mean here a sphere with some $n\geq 0$ handles, and below $S_n$ ...
9
votes
1answer
162 views

Does a pair of disjoint homotopic cycles in the dual separate the graph?

Let $G$ be a graph embedded on an orientable compact surface of genus $g$ so that the embedding is cellular. Consider the dual of the graph $G^*$. Let $C_1$ and $C_2$ be disjoint cycles in $G^*$ that ...
5
votes
1answer
354 views

Place n points in a box as far away from each other as possible

Can you suggest an optimal or heuristic algorithm for placing points on a 2D plane (within a constrained space) such that minimum distance between any two points is maximized. In other words, I'm ...
6
votes
3answers
751 views

Is there any good and free Introduction to topological graph theory

My knowledge in topological graph theory is in low, I need some good reference which has two simple thing, Definition of new concepts (like genus,graph embedding in surface, ...) also contains related ...