Simultaneous embedding is concerned with simultaneously representing a series of graphs sharing some or all vertices. This forms the basis for the visualization of dynamic graphs and thus is an important field of research. Recently there has been a great deal of work investigating simultaneous embedding problems both from a theoretical and a practical point of view. We survey recent work on this topic.
A result not really fitting in one of the three above classes by @cite_7 considers the restricted case of SEFE where each edge has to be a sequence of horizontal and vertical segments with at most one bend per edge. They show that two graphs with maximum degree 2 always admit such a SEFE on a grid of size @math by adapting their linear-time algorithm computing an SGE for these types of graphs (on a larger grid).
@cite_7
81810990 2569784685 1521196338 2531021157
Given k planar graphs G1,…,Gk, deciding whether they admit a simultaneous embedding with fixed edges (Sefe ) and whether they admit a simultaneous geometric embedding (Sge ) are NP-hard problems, for k≥3 and for k≥2, respectively. In this paper we consider the complexity of Sefe and of Sge when the graphs G1,…,Gk have a fixed planar embedding. In sharp contrast with the NP-hardness of Sefe for three non-embedded graphs, we show that Sefe is polynomial-time solvable for three graphs with a fixed planar embedding. Furthermore, we show that, given k embedded planar graphs G1,…,Gk, deciding whether a Sefe of G1,…,Gk exists and deciding whether an Sge of G1,…,Gk exists are NP-hard problems, for k≥14 and k≥13, respectively. Abstract In this paper we continue the study of the edge intersection graphs of single bend paths on a rectangular grid (i.e., the edge intersection graphs where each vertex is represented by one of the following shapes: ⌞ , ⌜ , ⌟ , ⌝ ). These graphs, called B 1 - EPG graphs, were first introduced by (2009) [Golumbic, M. C., M. Lipshteyn and M. Stern, Edge intersection graphs of single bend paths on a grid , Networks 54 :3 (2009), 130–138]. We focus on the class [⌞] (the edge intersection graphs of ⌞-shapes) and show that testing for membership in [⌞] is NP-complete. We then give a characterization and polytime recognition algorithm for special subclasses of Split ∩ [ ⌞ ] . We also consider the natural subclasses of B 1 -EPG formed by the subsets of the four single bend shapes (i.e., ⌞ , ⌞ , ⌜ , ⌞ , ⌝ , ⌞ , ⌜ , ⌝ – note: all other subsets are isomorphic to these up to 90 degree rotation). We observe the expected strict inclusions and incomparability (i.e., [ ⌞ ] ⊊ [ ⌞ , ⌜ ] , [ ⌞ , ⌝ ] ⊊ [ ⌞ , ⌜ , ⌝ ] ⊊ B 1 -EPG and [ ⌞ , ⌜ ] is incomparable with [ ⌞ , ⌝ ] ). We introduce the following notion of compressing an undirected graph G with (nonnegative) edge-lengths and terminal vertices R⊆V(G). A distance-preserving minor is a minor G′ (of G) with possibly different edge-lengths, such that R⊆V(G′) and the shortest-path distance between every pair of terminals is exactly the same in G and in G′. We ask: what is the smallest f*(k) such that every graph G with k=|R| terminals admits a distance-preserving minor G′ with at most f*(k) vertices? Simple analysis shows that f*(k)≤O(k4). Our main result proves that f*(k)≥Ω(k2), significantly improving over the trivial f*(k)≥k. Our lower bound holds even for planar graphs G, in contrast to graphs G of constant treewidth, for which we prove that O(k) vertices suffice. In our LICS 2004 paper we introduced an approach to the study of the local structure of finite algebras and relational structures that aims at applications in the Constraint Satisfaction Problem (CSP). This approach involves a graph associated with an algebra @math or a relational structure A, whose vertices are the elements of @math (or A), the edges represent subsets of @math such that the restriction of some term operation of @math is ‘good’ on the subset, that is, act as an operation of one of the 3 types: semilattice, majority, or affine. In this paper we significantly refine and advance this approach. In particular, we prove certain connectivity and rectangularity properties of relations over algebras related to components of the graph connected by semilattice and affine edges. We also prove a result similar to 2-decomposition of relations invariant under a majority operation, only here we do not impose any restrictions on the relation. These results allow us to give a new, somewhat more intuitive proof of the bounded width theorem: the CSP over algebra @math has bounded width if and only if @math does not contain affine edges. Actually, this result shows that bounded width implies width (2,3). We also consider algebras with edges from a restricted set of types. In particular, it can be proved that type restrictions are preserved under the standard algebraic constructions. Finally, we prove that algebras without semilattice edges have few subalgebras of powers, that is, the CSP over such algebras is also polynomial time.