### TABLE 1. Complexity results for ten famous graph problems when restricted to the classes of graphs discussed in the text. The ten problems are INDEPENDENT SET, CLIQUE, PARTITION INTO CLIQUES, CHROMATIC NUMBER, CHROMATIC INDEX, HAMILTONIAN CIRCUIT, DOMINATING SET, SIMPLE (unweighted) MAX CUT, (unweighted) STEINER TREE IN GRAPHS, and GRAPH ISOMORPHISM. The first nine are known to be NP-complete for general graphs; the complexity of GRAPH ISOMORPHISM for general graphs is a long-standing open problem. The well-known VERTEX COVER problem is not included as its complexity will always be the same as that of INDEPENDENT SET; see [G amp;J, p.54]. The second column gives the com- plexity of determining membership in the given class (and constructing the associated model where appropriate, as in the case of intersection graphs). For a key to the abbreviations used as entries in the table, see the continuation of the table on the next page.

1985

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### Table 1: Hits by Major Section Index

"... In PAGE 3: ...206 Programming Language Index Hits Implementations C language 805 37 C++ 929 11 Fortran 125 6 Lisp 99 1 Mathematica 104 3 Pascal 272 5 Totals 2334 63 Table 2: Hits by Programming Language Index Most Popular Problems Hits Least Popular Problems Hits shortest-path 681 shape-similarity 156 traveling-salesman 665 factoring-integers 141 minimum-spanning-tree 652 independent-set 137 kd-trees 611 cryptography 136 nearest-neighbor 609 maintaining-arrangements 134 triangulations 600 text-compression 133 voronoi-diagrams 578 generating-subsets 133 convex-hull 538 set-packing 126 graph-data-structures 519 planar-drawing 120 sorting 485 median 118 string-matching 467 satis ability 116 dictionaries 459 bandwidth 107 geometric-primitives 452 shortest-common-superstring 105 topological-sorting 424 feedback-set 83 su x-trees 423 determinants 78 Table 3: Most and least popular algorithmic problems, by repository hits. Table1 reports the number of hits distributed among our highest level of classi cation { the seven major sub elds of algorithms. Two di erent hit measures are reported for each sub eld, rst the number of hits to the menu of problems within the sub eld, and second the total number of hits to individual problem pages within this sub eld.... ..."

### Table 1. assembly of grains graph index number of elements

"... In PAGE 21: ...2 Graph models of heterogeneous media Let us pursue the planar graph representation of granular media in some more detail, Satake @71,72#. First, we list in Table1 a correspondence between a system of rotund grains and its graph model. Besides the vertex and edge sets intro- duced earlier, we also have a loop set L.... ..."

Cited by 2

### Table 2. Bicriteria spanning tree results for treewidth-bounded graphs.

"... In PAGE 6: ... As before, the rows are indexed by the budgeted objective. All algorithmic results in Table2 also extend to Steiner trees in a straightforward way. Our results for treewidth-bounded graphs have an interesting application in the context of find- ing optimum broadcast schemes.... In PAGE 19: ...1 Exact Algorithms Theorem 8.1 Every problem in Table2 can be solved exactly in O((n C)O(1))-time for any class of treewidth bounded graphs with no more than k terminals, for fixed k and a budget C on the first objective. The above theorem states that there exist pseudopolynomial-time algorithms for all the bicriteria problems from Table 2 when restricted to the class of treewidth-bounded graphs.... In PAGE 19: ...1 Every problem in Table 2 can be solved exactly in O((n C)O(1))-time for any class of treewidth bounded graphs with no more than k terminals, for fixed k and a budget C on the first objective. The above theorem states that there exist pseudopolynomial-time algorithms for all the bicriteria problems from Table2 when restricted to the class of treewidth-bounded graphs. The basic idea is to employ a dynamic programming strategy.... In PAGE 23: ...7 For the class of treewidth-bounded graphs, there is an FPAS for the (Diame- ter, Total cost, Spanning tree)-bicriteria problem with performance guarantee (1; 1 + ). As mentioned before, similar theorems hold for the other problems in Table2 and all these results extend directly to Steiner trees. 8.... ..."

### Table 2. Bicriteria spanning tree results for treewidth-bounded graphs.

"... In PAGE 6: ... As before, the rows are indexed by the budgeted objective. All algorithmic results in Table2 also extend to Steiner trees in a straightforward way. Our results for treewidth-bounded graphs have an interesting application in the context of find- ing optimum broadcast schemes.... In PAGE 19: ...1 Exact Algorithms Theorem 8.1 Every problem in Table2 can be solved exactly in O((n C)O(1))-time for any class of treewidth bounded graphs with no more than k terminals, for fixed k and a budget C on the first objective. The above theorem states that there exist pseudopolynomial-time algorithms for all the bicriteria problems from Table 2 when restricted to the class of treewidth-bounded graphs.... In PAGE 19: ...1 Every problem in Table 2 can be solved exactly in O((n C)O(1))-time for any class of treewidth bounded graphs with no more than k terminals, for fixed k and a budget C on the first objective. The above theorem states that there exist pseudopolynomial-time algorithms for all the bicriteria problems from Table2 when restricted to the class of treewidth-bounded graphs. The basic idea is to employ a dynamic programming strategy.... In PAGE 23: ...7 For the class of treewidth-bounded graphs, there is an FPAS for the (Diame- ter, Total cost, Spanning tree)-bicriteria problem with performance guarantee (1; 1 + ). As mentioned before, similar theorems hold for the other problems in Table2 and all these results extend directly to Steiner trees. 8.... ..."

### Table 7 Maximum-2-plex problem on DIMACS graphs. Graph details

"... In PAGE 18: ... However, the upper limit on runtime was increased to 24hrs and MaxGlob- alCuts parameter was reduced to 50 for these runs. Table7 presents the outcome of this experiment. Columns G, n, m, d and !(G) provide the DI- MACS graph name, number of vertices, number of edges, edge density and clique number, respectively.... ..."

### Table 3: Index of access to explanation: easy problems

"... In PAGE 8: ...26 p lt; .001), see Table3 and hard prob- lems (F(1,94) = 36.60 p lt; .... ..."

### Table 1: Benchmarks on simple graph coloring problems.

1995

"... In PAGE 36: ... The rst-fail principle [17] is also used for labeling in the CHIP program. Table1 gives the timing recorded for both CHIP and PROCLANN to solve the simple graph- coloring problems from 10- to 250-vertices. They both manage to solve the problems in a timely manner.... ..."

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### Table 2: Benchmarks on hard graph coloring problems.

1995

"... In PAGE 36: ... PROCLANN requires, in general, less than half of the CHIP time to nd a solution in all the cases. Table2 records timing for four hard graph-coloring problems. CHIP cannot solve either of the problems with 125 vertices within 48 hours.... ..."

Cited by 13