Related papers: Testing Simultaneous Planarity when the Common Gra…
In this paper, we present a constructive and proof-relevant development of graph theory, including the notion of maps, their faces, and maps of graphs embedded in the sphere, in homotopy type theory. This allows us to provide an elementary…
We study spaces of realisations of linkages (weighted graphs) whose underlying graph is a series parallel graph. In particular, we describe an algorithm for determining whether or not such spaces are connected.
Our starting point is the observation that if graphs in a class C have low descriptive complexity in first order logic, then the isomorphism problem for C is solvable by a fast parallel algorithm (essentially, by a simple combinatorial…
Modern methods of graph theory describe a graph up to isomorphism, which makes it difficult to create mathematical models for visualizing graph drawings on a plane. The topological drawing of the planar part of a graph allows representing…
Let $\langle G_r,G_b \rangle$ be a pair of plane $st$-graphs with the same vertex set $V$. A simultaneous visibility representation with L-shapes of $\langle G_r,G_b \rangle$ is a pair of bar visibility representations…
We present a data structure that can maintain a simple planar graph under edge contractions in linear total time. The data structure supports adjacency queries and provides access to neighbor lists in $O(1)$ time. Moreover, it can report…
Bipartite graphs model the relationships between two disjoint sets of entities in several applications and are naturally drawn as 2-layer graph drawings. In such drawings, the two sets of entities (vertices) are placed on two parallel lines…
As two fundamental problems, graph cuts and graph matching have been investigated over decades, resulting in vast literature in these two topics respectively. However the way of jointly applying and solving graph cuts and matching receives…
A simple topological graph is $k$-quasiplanar ($k\geq 2$) if it contains no $k$ pairwise crossing edges, and $k$-planar if no edge is crossed more than $k$ times. In this paper, we explore the relationship between $k$-planarity and…
In this paper, we introduce the notion of 2-boundary planar graphs. A graph is 2-boundary planar if it has an embedding in the plane so that all vertices lie on the boundary of at most two faces and no edges are crossed. A proper coloring…
It is known that a graph isomorphism testing algorithm is polynomially equivalent to a detecting of a graph non-trivial automorphism algorithm. The polynomiality of the latter algorithm, is obtained by consideration of symmetry properties…
A graph with n vertices is 1-planar if it can be drawn in the plane such that each edge is crossed at most once, and is optimal if it has the maximum of 4n-8 edges. We show that optimal 1-planar graphs can be recognized in linear time. Our…
Vertex connectivity and edge connectivity are fundamental concepts in graph theory that have been widely studied from both structural and algorithmic perspectives. The focus of this paper is on computing these two parameters for graphs…
A (possibly denerate) drawing of a graph $G$ in the plane is approximable by an embedding if it can be turned into an embedding by an arbitrarily small perturbation. We show that testing, whether a straight-line drawing of a planar graph…
In this paper, we show that given a weighted, directed planar graph $G$, and any $\epsilon >0$, there exists a polynomial time and $O(n^{\frac{1}{2}+\epsilon})$ space algorithm that computes the shortest path between two fixed vertices in…
Motivated by applications to graph morphing, we consider the following \emph{compatible connectivity-augmentation problem}: We are given a labelled $n$-vertex planar graph, $\mathcal{G}$, that has $r\ge 2$ connected components, and $k\ge 2$…
The clustered planarity problem (c-planarity) asks whether a hierarchically clustered graph admits a planar drawing such that the clusters can be nicely represented by regions. We introduce the cd-tree data structure and give a new…
A planar orthogonal drawing of a planar 4-graph G (i.e., a planar graph with vertex-degree at most four) is a crossing-free drawing that maps each vertex of G to a distinct point of the plane and each edge of $G$ to a sequence of horizontal…
The problem Simultaneous Embedding with Fixed Edges (SEFE) asks for two planar graph $G^1 = (V^1, E^1)$ and $G^2 = (V^2, E^2)$ sharing a common subgraph $G = G^1 \cap G^2$ whether they admit planar drawings such that the common graph is…
The 2-blowup of a graph is obtained by replacing each vertex with two non-adjacent copies; a graph is biplanar if it is the union of two planar graphs. We disprove a conjecture of Gethner that 2-blowups of planar graphs are biplanar:…