β\beta-Stars or On Extending a Drawing of a Connected Subgraph

We consider the problem of extending the drawing of a subgraph of a given plane graph to a drawing of the entire graph using straight-line and polyline edges. We define the notion of star complexity of a polygon and show that a drawing $\Gamma_H$ of an induced connected subgraph $H$ can be extended with at most $\min\{ h/2, \beta + \log_2(h) + 1\}$ bends per edge, where $\beta$ is the largest star complexity of a face of $\Gamma_H$ and $h$ is the size of the largest face of $H$. This result significantly improves the previously known upper bound of $72|V(H)|$ [5] for the case where $H$ is connected. We also show that our bound is worst case optimal up to a small additive constant. Additionally, we provide an indication of complexity of the problem of testing whether a star-shaped inner face can be extended to a straight-line drawing of the graph; this is in contrast to the fact that the same problem is solvable in linear time for the case of star-shaped outer face [9] and convex inner face [13].Comment: Appears in the Proceedings of the 26th International Symposium on Graph Drawing and Network Visualization (GD 2018

Extending Partial Representations of Circle Graphs in Near-Linear Time

The partial representation extension problem generalizes the recognition problem for geometric intersection graphs. The input consists of a graph G, a subgraph $H ⊆ G$ and a representation $\mathcal{R}′$ of H . The question is whether G admits a representation $\mathcal{R}$ whose restriction to H is $\mathcal{R}′$. We study this question for circle graphs, which are intersection graphs of chords of a circle. Their representations are called chord diagrams. We show that for a graph with n vertices and m edges the partial representation extension problem can be solved in $O((n + m)α(n + m))$ time, thereby improving over an $O(n^3)$-time algorithm by Chaplick et al. (J Graph Theory 91(4), 365–394, 2019). The main technical contributions are a canonical way of orienting chord diagrams and a novel compact representation of the set of all canonically oriented chord diagrams that represent a given circle graph G, which is of independent interest

Properties of Random Graphs with Hidden Color

We investigate in some detail a recently suggested general class of ensembles of sparse undirected random graphs based on a hidden stub-coloring, with or without the restriction to nondegenerate graphs. The calculability of local and global structural properties of graphs from the resulting ensembles is demonstrated. Cluster size statistics are derived with generating function techniques, yielding a well-defined percolation threshold. Explicit rules are derived for the enumeration of small subgraphs. Duality and redundancy is discussed, and subclasses corresponding to commonly studied models are identified.Comment: 14 pages, LaTeX, no figure

Third case of the Cyclic Coloring Conjecture

The Cyclic Coloring Conjecture asserts that the vertices of every plane graph with maximum face size D can be colored using at most 3D/2 colors in such a way that no face is incident with two vertices of the same color. The Cyclic Coloring Conjecture has been proven only for two values of D: the case D=3 is equivalent to the Four Color Theorem and the case D=4 is equivalent to Borodin's Six Color Theorem, which says that every graph that can be drawn in the plane with each edge crossed by at most one other edge is 6-colorable. We prove the case D=6 of the conjecture

Advancements on SEFE and Partitioned Book Embedding Problems

In this work we investigate the complexity of some problems related to the {\em Simultaneous Embedding with Fixed Edges} (SEFE) of $k$ planar graphs and the PARTITIONED $k$-PAGE BOOK EMBEDDING (PBE-$k$) problems, which are known to be equivalent under certain conditions. While the computational complexity of SEFE for $k=2$ is still a central open question in Graph Drawing, the problem is NP-complete for $k \geq 3$ [Gassner {\em et al.}, WG '06], even if the intersection graph is the same for each pair of graphs ({\em sunflower intersection}) [Schaefer, JGAA (2013)]. We improve on these results by proving that SEFE with $k \geq 3$ and sunflower intersection is NP-complete even when the intersection graph is a tree and all the input graphs are biconnected. Also, we prove NP-completeness for $k \geq 3$ of problem PBE-$k$ and of problem PARTITIONED T-COHERENT $k$-PAGE BOOK EMBEDDING (PTBE-$k$) - that is the generalization of PBE-$k$ in which the ordering of the vertices on the spine is constrained by a tree $T$ - even when two input graphs are biconnected. Further, we provide a linear-time algorithm for PTBE-$k$ when $k-1$ pages are assigned a connected graph. Finally, we prove that the problem of maximizing the number of edges that are drawn the same in a SEFE of two graphs is NP-complete in several restricted settings ({\em optimization version of SEFE}, Open Problem $9$, Chapter $11$ of the Handbook of Graph Drawing and Visualization).Comment: 29 pages, 10 figures, extended version of 'On Some NP-complete SEFE Problems' (Eighth International Workshop on Algorithms and Computation, 2014

Extending Partial Representations of Circle Graphs in Near-Linear Time

The partial representation extension problem generalizes the recognition problem for geometric intersection graphs. The input consists of a graph G, a subgraph H ⊆ G and a representation H of H. The question is whether G admits a representation G whose restriction to H is H. We study this question for circle graphs, which are intersection graphs of chords of a circle. Their representations are called chord diagrams. We show that for a graph with n vertices and m edges the partial representation extension problem can be solved in O((n+m)α(n+m)) time, where α is the inverse Ackermann function. This improves over an O(n$^{3}$)-time algorithm by Chaplick, Fulek and Klavík [2019]. The main technical contributions are a canonical way of orienting chord diagrams and a novel compact representation of the set of all canonically oriented chord diagrams that represent a given circle graph G, which is of independent interest