Multiple-Source Shortest Paths in Embedded Graphs

Journal of Combinatorial Theory, Series B, 2001

Whitney's theorem states that 3-connected planar graphs admit essentially unique embeddings in the plane. We generalize this result to embeddings of graphs in arbitrary surfaces by showing that there is a function ξ : N 0 → N 0 such that every 3-connected graph admits at most ξ(g) combinatorially distinct embeddings of face-width ≥ 3 into surfaces whose Euler genus is at most g.

Lecture Notes in Computer Science, 1995

A planarizing set of a graph is a set of edges or vertices whose removal leaves a planar graph. It is shown that, if G is an n-vertex graph of maximum degree d and orientable genus g, then there exists a planarizing set of O( p dgn) edges. This result is tight within a constant factor. Similar results are obtained for planarizing vertex sets and for graphs embedded on nonorientable surfaces. Planarizing edge and vertex sets can be found in O(n + g) time, if an embedding of G on a surface of genus g is given.

Symposium on Discrete Algorithms, 2016

Given a planar graph G(V, E) and a partition of the neighbors of each vertex v ∈ V in four sets v, v, v, and v, the problem WINDROSE PLANARITY asks to decide whether G admits a windrose-planar drawing, that is, a planar drawing in which (i) each neighbor u ∈ v is above and to the right of v, (ii) each neighbor u ∈ v is above and to the left of v, (iii) each neighbor u ∈ v is below and to the left of v, (iv) each neighbor u ∈ v is below and to the right of v, and (v) edges are represented by curves that are monotone with respect to each axis. By exploiting both the horizontal and the vertical relationship among vertices, windrose-planar drawings allow to simultaneously visualize two partial orders defined by means of the edges of the graph. Although the problem is N P-hard in the general case, we give a polynomial-time algorithm for testing whether there exists a windrose-planar drawing that respects a combinatorial embedding that is given as part of the input. This algorithm is based on a characterization of the plane triangulations admitting a windrose-planar drawing. Furthermore, for any embedded graph admitting a windrose-planar drawing we show how to construct one with at most one bend per edge on an O(n) × O(n) grid. The latter result contrasts with the fact that straight-line windrose-planar drawings may require exponential area.

Proceedings of the fourtieth annual ACM symposium on Theory of computing - STOC 08, 2008

For every surface S (orientable or non-orientable), we give a linear time algorithm to test the graph isomorphism of two graphs, one of which admits an embedding of face-width at least 3 into S. This improves a previously known algorithm whose time complexity is n O(g) , where g is the genus of S. This is the first algorithm for which the degree of polynomial in the time complexity does not depend on g. The above result is based on two linear time algorithms, each of which solves a problem that is of independent interest. The first of these problems is the following one. Let S be a fixed surface. Given a graph G and an integer k ≥ 3, we want to find an embedding of G in S of facewidth at least k, or conclude that such an embedding does not exist. It is known that this problem is NP-hard when the surface is not fixed. Moreover, if there is an embedding, the algorithm can give all embeddings of face-width at least k, up to Whitney equivalence. Here, the face-width of an embedded graph G is the minimum number of points of G in which some non-contractible closed curve in the surface intersects the graph. In the proof of the above algorithm, we give a simpler proof and a better bound for the theorem by Mohar and Robertson concerning the number of polyhedral embeddings of 3-connected graphs.

Discrete Mathematics, 1994

We show that every 3-connected planar graph has a circular embedding in some nonspherical surface. More generally, we characterize those planar graphs that have a 2-representative embedding in some nonspherical surface.

Acta Mathematica Scientia, 2002

In this paper, the authors discuss the upper bound for the genus of strong embeddings for 3-connected planar graphs on higher surfaces. It is shown that the problem of determining the upper bound for the strong embedding of 3-connected planar neartriangulations on higher non-orientable surfaces is NP-hard. As a corollary, a theorem of Richter, Seymour and Siran about the strong embedding of 3-connected planar graphs is generalized to orientable surface.

Journal of Combinatorial Theory, Series B, 1984

European Journal of Combinatorics, 2017

A class of graphs that lies strictly between the classes of graphs of genus (at most) k − 1 and k is studied. For a fixed orientable surface S k of genus k, let A k xy be the minor-closed class of graphs with terminals x and y that either embed into S k−1 or admit an embedding Π into S k such that there is a Π-face where x and y appear twice in the alternating order. In this paper, the obstructions for the classes A k xy are studied. In particular, the complete list of obstructions for A 1 xy is presented.

Journal of Graph Algorithms and Applications, 2003

An upward embedding of an embedded planar graph specifies, for each vertex v, which edges are incident on v "above" or "below" and, in turn, induces an upward orientation of the edges from bottom to top. In this paper we characterize the set of all upward embeddings and orientations of an embedded planar graph by using a simple flow model, which is related to that described by Bousset [3] to characterize bipolar orientations. We take advantage of such a flow model to compute upward orientations with the minimum number of sources and sinks of 1-connected embedded planar graphs. We finally devise a new algorithm for computing visibility representations of 1-connected planar graphs using our theoretic results.

ACM-SIAM Symposium on Discrete Algorithms, 1990

We show that each plane graph of order n 2 3 has a straight line embedding on the n-2 by n-2 grid. This embedding is computable in time O(n). A nice feature of the vertex-coordinates is that they have a purely combinatorial meaning.

arXiv: Combinatorics, 2017

It is known that graphs cellularly embedded into surfaces are equivalent to ribbon graphs. In this work, we generalize this statement to broader classes of graphs and surfaces. Half-edge graphs extend abstract graphs and are useful in quantum field theory in physics. On the other hand, ribbon graphs with half-edges generalize ribbon graphs and appear in a different type of field theory emanating from matrix models. We then give a sense of embeddings of half-edge graphs in punctured surfaces and determine (minimal/maximal) conditions for an equivalence between these embeddings and half-edge ribbon graphs. Given some assumptions on the embedding, the geometric dual of a cellularly embedded half-edge graph is also identified.

Acta Mathematicae Applicatae Sinica, 1992

Proceedings of the 2021 7th Student Computer Science Research Conference (StuCoSReC), 2021

In this paper, we propose a convention for repre-senting non-planar graphs and their least-crossing embeddings in a canonical way. We achieve this by using state-of-the-art tools such as canonical labelling of graphs, Nauty’s Graph6 string and combinatorial representations for planar graphs. To the best of our knowledge, this has not been done before. Besides, we implement the men-tioned procedure in a SageMath language and compute embeddings for certain classes of cubic, vertex-transitive and general graphs. Our main contribution is an extension of one of the graph data sets hosted on MathDataHub, and towards extending the SageMath codebase.

Russian Mathematical Surveys, 2003

Journal of Combinatorial Theory, Series B, 1998

In a 1973 paper, Cooke obtained an upper bound on the possible connectivity of a graph embedded in a surface (orientable or nonorientable) of fixed genus. Furthermore, he claimed that for each orientable genus #>0 (respectively, nonorientable genus #Ä >0, #Ä {2) there is a complete graph of orientable genus # (respectively, nonorientable genus #Ä ) and having connectivity attaining his bound. It is false that there is a complete graph of genus # (respectively, nonorientable genus #Ä ), for every # (respectively #Ä ) and that is the starting point of the present paper. Ringel and Youngs did show that for each #>0 (respectively, #Ä >0, #Ä {2) there is a complete graph K n which embeds in S # (respectively N #Ä ) such that n is the chromatic number of surface S # (respectively, the chromatic number of surface N #Ä ). One then easily observes that the connectivity of this K n attains the upper bound found by Cook. This leads us to define two kinds of connectivity bound for each orientable (or nonorientable) surface. We define the maximum connectivity } max of the orientable surface S # to be the maximum connectivity of any graph embeddable in the surface and the genus connectivity } gen (S # ) of the surface to be the maximum connectivity of any graph which genus embeds in the surface. For nonorientable surfaces, the bounds } max (N #Ä ) and } gen (N #Ä ) are defined similarly. In this paper we first study the uniqueness of graphs possessing connectivity } max (S # ) or } max (N #Ä ). The remainder of the paper is devoted to the study of the spectrum of values of genera in the intervals [#(K n )+1, #(K n+1 )] and [#Ä (K n )+1, #Ä (K n+1 )] with respect to their genus and maximum connectivities.