### Tables 1 to 3 show the implemented embeddings for an arbitrary guest graph into

### Table 1. Performance Guarantees for finding spanning trees in an arbitrary graph on n nodes. Asterisks indicate results obtained in this paper. gt; 0 is a fixed accuracy parameter.

"... In PAGE 3: ... There he provides an approximation algorithm for the (Degree, Diameter, Spanning tree) problem with performance guarantee (O(log2 n); O(log n))6. The (Diameter, Total cost, Spanning tree) entry in Table1 corresponds to the diameter-constrained minimum spanning tree problem introduced earlier. It is known that this problem is NP-hard even in the special case where the two cost functions are identical [HL+89].... In PAGE 5: ... 3.1 General Graphs Table1 contains the performance guarantees of our approximation algorithms for finding spanning trees, S, under different pairs of minimization objectives, A and B. For each problem cataloged in the table, two different costs are specified on the edges of the undirected graph: the first objective is computed using the first cost function and the second objective, using the second cost function.... In PAGE 5: ... For example the entry in row A, column B, denotes the performance guarantee for the problem of minimizing objective B with a budget on the objective A. All the results in Table1 extend to finding Steiner trees with at most a constant factor worsening in the performance ratios. For the diagonal entries in the table the extension to Steiner trees follows from Theorem 6.... ..."

### Table 1. Performance Guarantees for finding spanning trees in an arbitrary graph on n nodes. Asterisks indicate results obtained in this paper. gt; 0 is a fixed accuracy parameter.

"... In PAGE 3: ... There he provides an approximation algorithm for the (Degree, Diameter, Spanning tree) problem with performance guarantee (O(log2 n); O(log n))6. The (Diameter, Total cost, Spanning tree) entry in Table1 corresponds to the diameter-constrained minimum spanning tree problem introduced earlier. It is known that this problem is NP-hard even in the special case where the two cost functions are identical [HL+89].... In PAGE 5: ... 3.1 General Graphs Table1 contains the performance guarantees of our approximation algorithms for finding spanning trees, S, under different pairs of minimization objectives, A and B. For each problem cataloged in the table, two different costs are specified on the edges of the undirected graph: the first objective is computed using the first cost function and the second objective, using the second cost function.... In PAGE 5: ... For example the entry in row A, column B, denotes the performance guarantee for the problem of minimizing objective B with a budget on the objective A. All the results in Table1 extend to finding Steiner trees with at most a constant factor worsening in the performance ratios. For the diagonal entries in the table the extension to Steiner trees follows from Theorem 6.... ..."

### Table 1: Summary of known results for minimal completions and deletions. The input is an arbitrary graph G = (V; E) whereas in the extraction columns the graph H = (V; E[F) is assumed to be given as a completion and the graph H = (V; E nD) as a deletion. The asterisk denotes that the corresponding result is obtained in this work, and the dash denotes that the combination is not meaningful. For graph classes that are not listed here, no results are known in any of the columns.

2007

"... In PAGE 20: ...f Y its label. This de nes uniquely the minimal chain completion H. Therefore all steps can be done in total time O(n + m). 7 Concluding remarks In Table1 we summarize our results by presenting them together with previously known results obtained for other graph classes. For each graph class we give whether the class is sandwich monotone, the best known running time of an algorithm for computing a minimal completion (MC) or a minimal deletion (MD) of an arbitrary graph into this class, and the best known running time of an algorithm for extracting minimal completions or deletions into this class from a given completion or deletion.... ..."

### Table 2: Random acyclic graphs

1995

"... In PAGE 24: ...esult (Theorem 5.5). As more assumptions are imposed on input structure, the behavior of pure parallelization degrades. For acyclic graphs it remains nonetheless very good, but far less so than for arbitrary graphs, as shown in Table2 . For instance, the best behavior obtained is a speed- up by a factor of about 2.... ..."

Cited by 4

### Table 1. Performance Guarantees for nding spanning trees in an arbitrary graph on n nodes. Asterisks indicate results obtained in this paper. gt; 0 is a xed accuracy parameter. The diagonal entries in the table follow as a corollary of a general result (Theorem 12.8) which is proved using a parametric search algorithm. The entry for (Degree, Degree, Spanning tree) follows by combining Theorem 12.8 with the O(log n)-approximation algorithm for the degree problem in [RM+93]. In [RM+93] they actually provide an O(logn)-approximation algorithm for the weighted degree problem. (The weighted degree of a subgraph is de ned as the maximum over all nodes of

"... In PAGE 3: ... Given the framework, it remains to reason and ll in the appropriate polynomial time subroutine that is applicable for the corresponding pair of objectives. Table1 contains the performance guarantees of our approximation algorithms for nding span- ning trees, S, under di erent pairs of minimization objectives, A and B. For each problem cataloged in the table, two di erent costs are speci ed on the edges of the undirected graph: the rst objective is computed using the rst cost function and the second objective, using the second cost function.... In PAGE 3: ... For example the entry in row A, column B, denotes the performance guarantee for the problem of minimizing objective B with a budget on the objective A. All the results in Table1 extend to nding Steiner trees with at most a constant factor worsening in the performance ratios. All the results in the table extend to nding Steiner trees with at most a constant factor worsening in the performance ratios (Exercise!).... In PAGE 4: ... There he provides an approximation algorithm for the (Degree, Diameter, Spanning tree) problem with performance guarantee (O(log2 n); O(log n))1. The (Diameter, Total cost, Spanning tree) entry in Table1 corresponds to the diameter- constrained minimum spanning tree problem introduced earlier. It is known that this problem is NP-hard even in the special case where the two cost functions are identical [HL+89].... ..."

### Table 3: Recognition results in % for the di erent graphs used for matching on the 22 gallery. Matching was done with phase. Recognition was done without phase.

"... In PAGE 3: ... As an alternative to the face bunch graph one can use an arbitrary model not in the gallery. Results are shown in Table3 . Performance consistently degrades in the order: individual models, face bunch graph, arbitrary model.... ..."

### Table 3: The DTD Graph Operations.

"... In PAGE 22: ... The DTD graph BZBC can then be converted to BZBCBC using Lemma 2. Theorem 2 Given two arbitrary DTD graphs BZ and BZBCBC, there is a finite sequence of DTD change operations shown in Table3 that can transform BZ to BZBCBC. Proof: The set of operations CUcreate-ver, add-edge, delete-verCV all have equivalent operations in the DTD change taxonomy.... ..."

### Table I: Task times and speedup parameters for three machines executing a linear sequence of three tasks. Task time k 1, but is otherwise arbitrary; communication time O = c. Proposition 2 For linear task graphs GP and mapping 0, we have SP (1 ? ) +

1994

Cited by 15

### Table I: Task times and speedup parameters for three machines executing a linear sequence of three tasks. Task time k 1, but is otherwise arbitrary; communication time O = c. Proposition 2 For linear task graphs GP and mapping 0, we have SP (1 ? ) +

1994

Cited by 15