### Table 3: Random Graphs

1994

"... In PAGE 8: ... If we de ne the den- sity of a graph G as the number of edges of G = (V; E) over the number of edges of the complete graph with jV j nodes, then for this class of random graphs the density is very close to p. In Table3 we compare our algorithm (column `MS apos;) with the fastest algorithms known so far, due to Carraghan and Pardalos [9], Balas and Xue [26], Babel [2]. The rst two alghorithms were run on a SUN4/260 workstation, and the results are shown in the columns labelled by `CP apos; and `BX apos;.... ..."

Cited by 11

### Table 1 Results for the complete graph C5.

2007

### Table 1 shows the number of iteration steps and the average convergence rate for di erent mesh sizes and di erent distributions of the unknowns on the processors. For instance, 128 1 is a distribution into 128 strips, and 16 8 refers to a partitioning into 16 8 quadrilaterals. The convergence rates are fairly independent of the mesh size h and the partitioning for the parallelization. Similar results have been obtained for discretizations on non-uniform meshes.

### Table 9 Performance of kPaToH with and without bipartite graph matching (BGM) in par- titioning hypergraphs with a randomly selected number (F ) of fixed vertices Avg. cutsize %Improvement %Share of

"... In PAGE 23: ... In general, compared to PaToH, the relative performance of kPaToH in minimizing the cutsize is better in BF and OP datasets, which have high variation in net sizes. In Table9 , we report performance results in terms of the cutsize on a subset of the hypergraphs selected from Table 1. In each hypergraph, we randomly fix F = 128, 256, 512 vertices to K = 128, 256, 512 parts in a round-robin fashion and partition the hypergraph using both PaToH and kPaToH.... In PAGE 23: ... For kPaToH, we explore the percent improvement due to bipartite graph matching (BGM) and hence try it both with and without BGM. In Table9 , kPaToH w/ BGM corresponds to an implementation of the approach described in Section 4, whereas kPaToH w/o BGM corresponds to an implementation in which the fixed vertex sets are arbitrarily matched with the ordinary vertex parts. Basically, kPaToH provides superior results over PaToH in partitioning hy- pergraphs with fixed vertices due to (i) K-way refinement and (ii) BGM performed during the the initial partitioning phase.... In PAGE 23: ... We provide Table 9 to illustrate the share of these two factors in the overall cutsize improvement. According to Table9 , the share of BGM in the total cutsize improvement is quite variable, ranging between 4.... In PAGE 25: ...Averages of the results provided in Table9 over F and K Averages over F Averages over K %Improvement %Share of %Improvement %Share of kPaToH kPaToH BGM in the kPaToH kPaToH BGM in the F w/o BGM w/ BGM improvement K w/o BGM w/ BGM improvement 256 9.57 14.... ..."

### Table 1: Separating examples between chordal probe graphs and related incomparable classes.

2005

### Table 3: Separating examples between k-EPT graphs and related incomparable classes.

2005

### Table 1: AS Graph Statistics

2002

"... In PAGE 2: ... in the Oregon-based AS graphs [3]. We compare the Oregon-based AS graph ( Table1 , first row) against our more complete AS graph (Table 1, last row). In Figure 2 we plot the complementary cumulative distribu- tion function of the AS degree.... In PAGE 2: ... in the Oregon-based AS graphs [3]. We compare the Oregon-based AS graph (Table 1, first row) against our more complete AS graph ( Table1 , last row). In Figure 2 we plot the complementary cumulative distribu- tion function of the AS degree.... ..."

Cited by 89

### Table 6: Numerical results on SDP relaxations of the graph partition problems. standard conversion completion

2003

"... In PAGE 19: ... Although (13) involves a dense data matrix E, we can obtain an equivalent SDP with sparse aggregate sparsity pattern applying an appropriate congruent transformation to it [8, section 6]. Table6 compares the three methods for the transformed problems. As k1 becomes large, the aggregate sparsity patterns remain sparse, though the extended sparsity patterns become dense for them.... ..."

Cited by 16

### Table 2: Comparing regular and irregular graphs.

2001

"... In PAGE 6: ... 4.1 Binary Symmetric Channel Table2 compares the performance of regular and irregular codes of rates 1/2 and 1/4. Our results for regular codes (based on graphs in which all nodes on the left have degree 3) are slightly better than (but consistent with) previous results reported in [11].... ..."

Cited by 103

### Table 2: Comparing regular and irregular graphs.

2001

"... In PAGE 6: ... 4.1 Binary Symmetric Channel Table2 compares the performance of regular and irregular codes of rates 1/2 and 1/4. Our results for regular codes (based on graphs in which all nodes on the left have degree 3) are slightly better than (but consistent with) previous results reported in [11].... ..."

Cited by 103