Single pushout graph transformations at injective matches are extended by negative application conditions. It is shown, how right-sided application conditions can be transformed into equivalent left-sided ones. Based on this result, conditional derived rules are introduced and a theorem, similiar to the double pushout embedding theorem is shown. Finally, three variants of contextual application...

We prove that we can specify by formulas of monadic second-order logic the unique planar embedding of a 3-connected planar graph. If the planar graph is not 3-connected but given with a linear order of its set of edges, we can also define a planar embedding by monadic second-order formulas. We cannot do so in general without the ordering, even for 2-connected planar graphs. The planar embedding...

The strong embedding conjecture states that every 2-connected graph has a closed 2-cell embedding in some surface , i . e . an embedding that each face is bounded by a circuit in the graph . A graph is called k -crosscap embeddable if it can be embedded in the surface of non-orientable genus k . We confirm the strong embedding conjecture for 5-crosscap embeddable graphs . As a corollary , every...

In this paper, we present a one-to-one embedding of a graph with bounded treewidth into its optimal hypercube. This is the first time that embeddings of graphs with a very irregular structure into hypercubes are investigated. The dilation of the presented embedding is bounded by , where denotes the treewidth of the graph and denotes the maximal degree of a vertex in the graph. The given embeddi...

We present a data structure for maintaining a planar embedding of a dynamic plane graph. The graph may be updated with edge deletions and insertions of edges when it does not violate the embedding. Queries of vertex pairs return the list of possible places for inserting an edge. Furthermore, the embedding may be updated with flips that alter the embedding by swapping the orientation of a subgra...

We show that every n-vertex planar graph admits a simultaneous embedding without mapping and with fixed edges with any (n/2)vertex planar graph. In order to achieve this result, we prove that every n-vertex plane graph has an induced outerplane subgraph containing at least n/2 vertices. Also, we show that every n-vertex planar graph and every n-vertex planar partial 3-tree admit a simultaneous ...

The geometric simultaneous embedding problem asks whether two planar graphs on the same set of vertices in the plane can be drawn using straight lines, such that each graph is plane. Geometric simultaneous embedding is a current topic in graph drawing and positive and negative results are known for various classes of graphs. So far only connected graphs have been considered. In this paper we pr...

We show that every n-vertex planar graph admits a simultaneous embedding with no mapping and with fixed edges with any (n/2)-vertex planar graph. In order to achieve this result, we prove that every n-vertex plane graph has an induced outerplane subgraph containing at least n/2 vertices. Also, we show that every n-vertex planar graph and every n-vertex planar partial 3-tree admit a simultaneous...