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We consider the following problem known as simultaneous geometric graph embedding (SGE). Given a set of planar graphs on a shared vertex set, decide whether the vertices can be placed in the plane in such a way that for each graph the straight-line drawing is planar. We partially settle an open problem of Erten and Kobourov [5] by showing that even for two graphs the problem is NP-hard. We also show that the problem of computing the rectilinear crossing number of a graph can be reduced to a simultaneous geometric graph embedding problem; this implies that placing SGE in NP will be hard, since the corresponding question for rectilinear crossing number is a long-standing open problem. However, rather like rectilinear crossing number, SGE can be decided in PSPACE. [ABSTRACT FROM AUTHOR]

We study the problem of simultaneously embedding several graphs on the same vertex set in such a way that edges common to two or more graphs are represented by the same curve. This problem is known as simultaneously embedding graphs with fixed edges. We show that this problem is closely related to the weak realizability problem: Can a graph be drawn such that all edge crossings occur in a given set of edge pairs? By exploiting this relationship we can explain why the simultaneous embedding problem is challenging, both from a computational and a combinatorial point of view. More precisely, we prove that simultaneously embedding graphs with fixed edges is NP-complete even for three planar graphs. For two planar graphs the complexity status is still open. [ABSTRACT FROM AUTHOR]

In the problem of Online Call Admission in Optical Networks, briefly called oca, we are given a graph G=(V,E) together with a set of wavelengths W (χ:=W) and a finite sequence σ=r1,r2,... of calls which arrive in an online fashion. Each call rj specifies a pair of nodes to be connected. A lightpath is a path in G together with a wavelength λ ∈ W. Upon arrival of a call, an online algorithm must decide immediately and irrevocably whether to accept or to reject the call without any knowledge of calls which appear later in the sequence. If the call is accepted, the algorithm must provide a lightpath to connect the specified nodes. The essential restriction is the wavelength conflict constraint: each wavelength is available only once per edge, which implies that two lightpaths sharing an edge must have different wavelengths. The objective in oca is to maximize the overall profit, that is, the number of accepted calls. A result by Awerbuch et al. states that a c-competitive algorithm for oca with one wavelength, i.e., χ:=W=1, implies a (c+1)-competitive algorithm for general numbers of wavelengths. However, for instance, for the line with n+1 nodes, a lower bound of n for the competitive ratio of deterministic algorithms for χ=1 makes this result void in many cases. We provide a deterministic competitive algorithm for wavelengths which achieves a competitive ratio of on the line with n+1 nodes. As long as is fixed, this is the first competitive ratio which is sublinear in n+1, the number of nodes. [ABSTRACT FROM AUTHOR]