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Two planar graphs G 1 and G 2 sharing some vertices and edges are simultaneously planar if they have planar drawings such that a shared vertex [edge] is represented by the same point [curve] in both drawings. It is an open problem whether simultaneous planarity can be tested efficiently. We give a linear-time algorithm to test simultaneous planarity when the two graphs share a 2-connected subgraph. Our algorithm extends to the case of k planar graphs where each vertex [edge] is either common to all graphs or belongs to exactly one of them, and the common subgraph is 2-connected.

Abstract. One advantage of smart grids is that they can reduce the peak load by distributing electricity-demands over multiple short intervals. Finding a schedule that minimizes the peak load corresponds to a variant of a strip packing problem. Normally, for strip packing problems, a given set of axis-aligned rectangles must be packed into a fixed-width strip, and the goal is to minimize the height of the strip. The electricityallocation application can be modelled as strip packing with slicing: each rectangle may be cut vertically into multiple slices and the slices may be packed into the strip as individual pieces. The stacking constraint forbids solutions in which a vertical line intersects two slices of the same rectangle. We give a fully polynomial time approximation scheme for this problem, as well as a practical polynomial time algorithm that slices each rectangle at most once and yields a solution of height at most 5/3 times the optimal height.

We introduce a notion of simultaneity for any class of graphs with an intersection representation (interval graphs, chordal graphs, etc.) and for comparability graphs, which are represented by transitive orientations. Let G 1 and G 2 be graphs from such a class C, sharing some vertices I and the corresponding induced edges. Then G 1 and G 2 are said to be simultaneous C graphs if there exist representations R 1 and R 2 of G 1 and G 2 that are the same on the shared subgraph.Simultaneous representation problems arise in any situation where two related graphs should be represented consistently. A main instance is for temporal relationships, where an old graph and a new graph share some common parts. Pairs of related graphs arise in many other situations. For example, two social networks that share some members; two schedules that share some events, overlap graphs of DNA fragments of two similar organisms, circuit graphs of two adjacent layers on a computer chip, etc.For comparability graphs and any intersection graph class, we show that the simultaneous representation problem for the graph class is equivalent to a graph augmentation problem: given graphs G 1 and G 2 , sharing vertices I and the corresponding induced edges, do there exist edges E ⊆ V (G 1 ) − I × V (G 2 ) − I such that the graph G 1 ∪ G 2 ∪ E belongs to the graph class. This equivalence implies that the simultaneous representation problem is closely related to some well-studied classes in the literature, namely, sandwich graphs and probe graphs.We give efficient algorithms for solving the simultaneous representation problem for chordal graphs, comparability graphs and permutation graphs. Further, our algorithms for comparability and permutation graphs solve a more general version of the problem when there are multiple graphs, any two of which share the same common graph. This version of the problem also generalizes probe graphs. Finally, we show that the general version of the problem is NP-hard for chordal graphs.