Characterizing Simultaneous Embeddings with Fixed Edges

A set of planar graphs share a simultaneous embedding if they can be drawn on the same vertex set V in the plane without crossings between edges of the same graph. Fixed edges are common edges between graphs that share the same Jordan curve in the simultaneous drawings. While any number of planar graphs have a simultaneous embedding without fixed edges, determining which graphs always share a simultaneous embedding with fixed edges (SEFE) has been open. We partially close this problem by giving a necessary condition to determine when pairs of graphs have a SEFE

Characterizations of Restricted Pairs of Planar Graphs allowing Simultaneous Embeddings with Fixed Edges

A set of planar graphs share a simultaneous embedding if they can be drawn on the same vertex set V in the Euclidean plane without crossings between edges of the same graph. Fixed edges are common edges between graphs that share the same simple curve in the simultaneous drawing. Determining in polynomial time which pairs of graphs share a simultaneous embedding with ?xed edges (SEFE) has been open. We give a necessary and su?cient condition for whether a SEFE exists for pairs of graphs whose union is homeomorphic to K5 or K3,3 . This allows us to characterize the class of planar graphs that always have a SEFE with any other planar graph. We also characterize the class of biconnected outerplanar graphs that always have a SEFE with any other outerplanar graph. In both cases, we provide e?cient algorithms to compute a SEFE. Finally, we provide a linear-time decision algorithm for deciding whether a pair of biconnected outerplanar graphs has a SEFE

Characterizing Simultaneous Embeddings with Fixed Edges

A set of planar graphs share a simultaneous embedding if they can be drawn on the same vertex set V in the plane without crossings between edges of the same graph. Fixed edges are common edges between graphs that share the same Jordan curve in the simultaneous drawings. While any number of planar graphs have a simultaneous embedding without ?xed edges, determining which graphs always share a simultaneous embedding with ?xed edges (SEFE) has been open. We partially close this problem by giving a necessary condition to determine when pairs of graphs have a SEFE

An SPQR-Tree Approach to Decide Special Cases of Simultaneous Embedding with Fixed Edges

We present a linear-time algorithm for solving the simulta- neous embedding problem with ?xed edges (SEFE) for a planar graph and a pseudoforest (a graph with at most one cycle) by reducing it to the following embedding problem: Given a planar graph G, a cycle C of G, and a partitioning of the remaining vertices of G, does there exist a planar embedding in which the induced subgraph on each vertex partite of G C is contained entirely inside or outside C ? For the latter prob- lem, we present an algorithm that is based on SPQR-trees and has linear running time. We also show how we can employ SPQR-trees to decide SEFE for two planar graphs where one graph has at most two cycles and the intersection is a pseudoforest in linear time. These results give rise to our hope that our SPQR-tree approach might eventually lead to a polynomial-time algorithm for deciding the general SEFE problem for two planar graphs

Automated Cryocooler Monitor and Control System

A system was designed to automate cryogenically cooled low-noise amplifier systems used in the NASA Deep Space Network. It automates the entire operation of the system including cool-down, warm-up, and performance monitoring. The system is based on a single-board computer with custom software and hardware to monitor and control the cryogenic operation of the system. The system provides local display and control, and can be operated remotely via a Web interface. The system controller is based on a commercial single-board computer with onboard data acquisition capability. The commercial hardware includes a microprocessor, an LCD (liquid crystal display), seven LED (light emitting diode) displays, a seven-key keypad, an Ethernet interface, 40 digital I/O (input/output) ports, 11 A/D (analog to digital) inputs, four D/A (digital to analog) outputs, and an external relay board to control the high-current devices. The temperature sensors used are commercial silicon diode devices that provide a non-linear voltage output proportional to temperature. The devices are excited with a 10-microamp bias current. The system is capable of monitoring and displaying three temperatures. The vacuum sensors are commercial thermistor devices. The output of the sensors is a non-linear voltage proportional to vacuum pressure in the 1-Torr to 1-millitorr range. Two sensors are used. One measures the vacuum pressure in the cryocooler and the other the pressure at the input to the vacuum pump. The helium pressure sensor is a commercial device that provides a linear voltage output from 1 to 5 volts, corresponding to a gas pressure from 0 to 3.5 MPa (approx. = 500 psig). Control of the vacuum process is accomplished with a commercial electrically operated solenoid valve. A commercial motor starter is used to control the input power of the compressor. The warm-up heaters are commercial power resistors sized to provide the appropriate power for the thermal mass of the particular system, and typically provide 50 watts of heat. There are four basic operating modes. "Cool " mode commands the system to cool to normal operating temperature. "Heat " mode is used to warm the device to a set temperature near room temperature. "Pump " mode is a maintenance function that allows the vacuum system to be operated alone to remove accumulated contaminants from the vacuum area. In "Off " mode, no power is applied to the system