Index. C cmp;l, 126. C rq;l, 127. A A sp. B BANDWIDTH, 32, 218 complexity of, 32 k-bandwidth, 32, 218
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1 , 24, 96, 116 -compatible, , 38 [], 125 P;l, 25 Q;l, 127 A A sp I, 166, 173 A sp R, 166, A tw I, 189, 205 A tw R, 189, adjacency list representation, 13 adjacent, 9 algorithm, reduction, see reduction algorithm analysis parallel algorithm, 13 sequential algorithm, 13 B BANDWIDTH, 32, 218 complexity of, 32 bandwidth, 31, 85 k-bandwidth, 32, 218 complexity of, 32 biconnected, 10 biconnected component, 10 block, 10 connecting, 66 non-connecting, 66 non-trivial, 10 pseudo, see pseudo block sandwich, 75 trivial, 10 underlying, 193 block state, 57 boundary, 24 bounded adjacency list method, 99 branchwidth, 4 bridge, 10, 189 C C cmp;l, 126 C Q;l, 127 C rq;l, 127 cell completion, 38 child, 11 Cholesky factorization, 4 chord, 11 chordal, 11, 30 CHROMATIC NUMBER, 138, 145, 218 clique, 11 COLORABILITY, 218 k-colorability, 27, 121, 131, 218 the constructive version of, 27 coloring, 72 k-coloring, 218 compatible, 126 complete, 97, 109 component, 10 biconnected, 10 connected, 10 concatenation (++), 38 cond(st), 57 cond 1 (st), 63 conflict graph, 114 connected, 10 construction problem, see constructive decision problem construction property, 122 derived, 132 constructive decision problem,
2 algorithm for, 12 MS-definable, 27 reduction algorithm for, constructive optimization problem, 12 algorithm for, 12 MS-definable, 28 reduction algorithm for, 133 constructive reduction algorithm, 122, 124, 133 constructive reduction algorithms, for decision problems, , for multigraphs, 141 for optimization problems, , parallel, sequential, constructive reduction system, 122 derived, 132 special, 123 special parallel, 138 constructive reduction-counter system, 131 special, 132 special parallel, 140 contraction, 11 control-flow graph, 3 correct cycle path, 54 COVERING BY CLIQUES, 219 CRCW PRAM, 13 cut, 218 cut vertex, 10 strong, 193 cycle, 10 chord of, 11 chordless, 11 cycle path, 40 correct, 54 cycle-sequence, 177 bounding paths of, 177 D D [], 125 dangling edge, 192 bad, 202 decision problem, 12 algorithm for, 12 see constructive decision problem MS-definable, 26 non- 12 reduction algorithm for, decreasing, 98, 109 deg, 9 degree, 9 degree constraints, 168 descendant, 10 diagonal, 90 d-discoverable, 99, 117 distance, 10 disturb, 195 DLSPG, 161 DNA physical mapping, 4, 71 DOMINATING SET, 218 p-dominating SET, 218 DSPG, 161 dst i (u;v),56 dynamic programming, 2, E E1, 52 E2, 52 ec Q;l, 127 ec rq;l, 127 edge, 9 bad, 180, 202 dangling, see dangling edge end, see end edge end points of, 9 multiple, 10 parallel, 10 edge contraction, 11 effectively decidable, 96 end edge,
3 double, 37 end point, 9 end vertex, 37 double, 37 ending point, 79 equivalence relation, 25, 127 refinement, 104 EREW PRAM, 13 expert system, 3 extension ( z), 134 extension constants (d l ), 134 F finite index, 25 finite integer index, 110 finite state, 23, 25 fixed parameter tractable, 32, 74 forbidden minors, minimal, 28 forest, 10 G G empty,98 gate matrix layout, 4 graph, 9 11 bandwidth of, 31 biconnected, 10 boundaried, 24 chordal, 11, 30 clique in, 11 complete, 11 conflict, 114 connected, 10 control-flow, 3 directed, 9 induced, 10 interval, see interval graph isomorphic, 11 layout of, 31 multi-, see multigraph path decomposition of, 14 pathwidth of, 14 proper path decomposition of, 85 proper pathwidth of, 85 sandwich, see sandich graph series-parallel, 33 simple, 9 source-sink labeled, 32 sourced, 24 terminal, see terminal graph tree decomposition of, 13 treewidth of, 14 triangulated, 11 underlying, 75 graph class, 12 cutset regular, 25 finite state, 25 fully cutset regular, 25 minor-closed, 28 MS-definable, 26 obstruction set of, 29 recognizable, 25 regular, 25 graph optimization problem, see optimization problem graph problem, see problem graph problems, 11 13, graph property, 24 derived, 109 effectively decidable, 96 extended, 24 finite index, 25 MS-definable, 26 H HAMILTONIAN CIRCUIT, 12, 27, 131, 217 constructive version of, 27 Hamiltonian circuit, 217 HAMILTONIAN CIRCUIT COMPLETION,
4 HAMILTONIAN PATH, 218 Hamiltonian path, 217 HAMILTONIAN PATH COMPLETION, 219 I I sp, 166, 168 I tw, 190 I1, 52 I2, 52 ICG, 73, ICG, 81 k-icg, 220 incident, 9 INDEPENDENT SET, 12, 218 independent set, 1, 218 k-independent SET,13 INDUCED d-degree SUBGRAPH, 218 induced graph, 10 induced subgraph, 10 inducible, 125 interval completion, 31, 85 interval graph, 30 unit, 73 interval realization, 30 interval routing, 4 intervalization, 72, 73 k-intervalization, 72, 73 INTERVALIZING COLORED GRAPHS, see ICG INTERVALIZING SANDWICH GRAPHS, see ISG intervalizing sandwich graphs, irreducible, 97 ISG, 73, ISG, 76 3-ISG, ISG, k-isg, 220 isomorphic, 11 isomorphism, 11 J join-reduce round, 114 L LARGE CUT, 219 layout, 31, 85 legal, 85 leaf, 11 leaf node, 11, 33 LEAF SPANNING TREE, 219 level, 11 LONG CYCLE, 219 LONG PATH, 219 LONGEST CYCLE, 28, 155, 219 constructive version of, 28 LONGEST PATH, 28, 155, 219 constructive version of, 28 LSPG, 161 LSPG reduction system for, 167 M match, 97, 108, 116, 168, 190 d-discoverable, 99, 117 disturbed, 194 non-disturbed, 194 matches non-interfering, 112 MAX CUT, 28, 112, 116, 138, 141, 144, 155, 219 constructive version of, 28 MAX INDEPENDENT SET, 12, 21, 28, 108, 111, 121, 131, 218 constructive version of, 28 MAX INDEPENDENT SET on cycles, 109, 110, 132, 140 MAX INDUCED d-degree SUBGRAPH, 111, 116, 138, 141, 144, 218 MAX LEAF SPANNING TREE, 112, 116, 138, 141, 145, 219 maximum independent set, 1 MIN BANDWIDTH, 218 MIN COVERING BY CLIQUES, 155, 219 MIN DOMINATING SET, 218 MIN p-dominating SET, 112, 116, 138, 242
5 141, 144, 218 MIN HAMILTONIAN CIRCUIT COMPLETION, 145, 219 MIN HAMILTONIAN PATH COMPLETION, 112, 116, 138, 145, 219 MIN PARTITION INTO CLIQUES, 112, 116, 144, 219 MIN PATHWIDTH, 19, 217 MIN TREEWIDTH, 19, 217 MIN VERTEX COVER, 111, 116, 138, 141, 218 minor, 11 forbidden, see forbidden minors minor-closed, 28 Monadic Second Order Logic, 24, MS-definable, MSOL, multigraph, 9 B-labeled, 189 terminal, see terminal multigraph N N,52 natural language processing, 3 neighbor, 9 node, 14 child, 11 leaf, 11 non-interfering, 112, 138 O obstruction set, 29 occur, 37 occurrence, 37 operations, 13 opt, 131 optimal speedup, 13 optimization problem, 12 algorithm for, 12 see constructive optimization problem MS-definable, 27 non- 12 reduction algorithm for, opts, 134 overlap information non-negative, 71 positive, 71 P P,26 P G,66 P H,50 P H,51 P k (H),48 P k (H),48 p-node, 33 parallel composition, 33 parallel node, 33 parallel reduction, 35, 95 partial k-path, 15 partial solution, 125 partial k-tree, 15 partial two-paths, biconnected, sequential algorithm for, structure of, trees, PARTITION INTO CLIQUES, 138, 219 path, 10, 51, 66 cycle, 40 path decomposition, 3, 14, 75 proper, 85 properties of, path of cycles, 39 k-path, 15 PATHWIDTH, 19, 217 complexity of, 19 pathwidth, 2, 14, 75 proper, 85 properties of, pathwidth two trees of,
6 2-PATHWIDTH parallel algorithm for, 212 sequential algorithm for, k-pathwidth, 19, 217 algorithms for, 19 complexity of, 19 PB, 193 perfect matching, 122 perfect phylogeny, 4 PRAM, 13 predicate, 26 MSOL, 26 problem, 12 construction, see construction problem decision, see decision problem graph, 11 graph optimization, 108 optimization, 12 real-life, 1 recognition, 12 pseudo block, 192 degree d, 193 pseudo block tree, 193 pw, 14 PW2, 79 R R sp, 166, 167 R tw, 189, 190, 191 RAM, 13 recognition problem, 12 Reduce, 101 Reduce-Construct, 124 reduction, 97, 108 parallel, 35, 95 series, 35, 95 reduction algorithm, 6, 28, 95, 101, 109 see constructive reduction algorithm efficient, 101, 110 parallel, 113, 115 reduction algorithms, applications of, for decision problems, , for multigraphs, for optimization problems, , parallel, sequential, reduction rule, 95, 97, 116 application of, 97, 116 match to, 97, 168, 190 safe, 97 reduction rules complete, 97 decreasing, 98 safe for LSPG, safe for TW2, terminating, 97 reduction system, 98 see constructive reduction system decreasing, 98 derived, 109 special, 100, 117 special for multigraphs, 117 special parallel, 112, 117 special parallel for multigraphs, 117 reduction systems, reduction-counter rule, 96, 108 application of, 108 match to, 108 safe, 108 reduction-counter rules complete, 109 decreasing, 109 terminating, 109 reduction-counter system, 109 see constructive reduction-counter system special,
7 special for multigraphs, 118 special parallel, 115 refinement, 104 register allocation, 3 root, 10 S S, 193 S,52 S sp, 166 S tw, 189 s-node, 33 safe, 97, 108 SANDWICH BANDWIDTH, 86, 220 k-sandwich BANDWIDTH, 220 sandwich block, 75 sandwich graph, 72 bandwidth of, 85 layout of, 85 legal layout of, 85 path decomposition of, 75 pathwidth of, 75 proper path decomposition of, 85 proper pathwidth of, 85 SANDWICH PATHWIDTH, 76, 220 k-sandwich PATHWIDTH, 220 SANDWICH PROPER PATHWIDTH, 86, 220 k-sandwich PROPER PATHWIDTH, 220 separator, 10, 204 x;y-separator, 204 minimal, 204 sequence reconstruction, 71 series composition, 32 series node, 33 series reduction, 35, 95 SERIES-PARALLEL GRAPH, 161, 220 series-parallel graph, 33 base, 33, 166 sp-tree of, 33 series-parallel graphs parallel algorithm for, reduction system for, 167 sequential algorithm for, 35 sink, 32 solution, 121, 122 partial, 125 solution domain, 122 inducible, 125 partial, 125 source, 24, 32 SOURCE-SINK LABELED SERIES- PARALLEL GRAPH, 161, 220 sp-tree, 33 binary, 34 minimal, 34 spanning tree, 219 special parallel reduction system, 112, 117 for multigraphs, 117 special reduction system, 100, 117 for multigraphs, 117 special reduction-counter system, 118 for multigraphs, 118 SPG, 161 star, 192 starting point, 79 state block, 57 vertex, 52 subgraph, 10 supergraph, 10 T telephone network, 4 terminal, 24 terminal graph, 24, 96, 116 d-discoverable, 99, 117 isomorphic, 96 open, 24 terminal multigraph, 116 B-labeled, 189 terminating, 97, 109 THREE-PARTITION,82 traveling salesman problem, 1, 4 tree,
8 depth of, 11 pseudo block, 193 root of, 10 rooted, 10 rooted binary, 11 tree decomposition, 2, 13, 189 dynamic programming on, node in, 14 properties of, rooted binary, 17 special, 204 width of, 14 tree of cycles, 39 k-tree, 15 trees of a graph, 50 TREEWIDTH, 19, 217 complexity of, 19 treewidth, 2, 14 properties of, TREEWIDTH AT MOST TWO, 190, TREEWIDTH, TREEWIDTH reduction system for, TREEWIDTH, 187, 189 parallel algorithm for, k-treewidth, 19, 217 algorithms for, 19 complexity of, 19 triangulated graph, 11 triangulation, 30 TW2, 190, 221 reduction system for, 191 tw, 14 two-colorability, 113 underlying graph, 75 unit interval graph, 73 unit-intervalization, 74 k-unit-intervalization, 74 UNIT-INTERVALIZING COLORED GRAPHS, see UICG UNIT-INTERVALIZING SANDWICH GRAPHS, see UISG unit-intervalizing sandwich graphs, V vertex, 9 boundary, 24 cut, see cut vertex degree of, 9 descendants of, 10 end, see end vertex inner, 24 internal, 11 level of, 11 neighbor of, 9 source, 24 terminal, 24 VERTEX COVER, 144, 218 vertex state, 52 t-vertex-edge-tuple, 129 W W [i], 32 walk, 10 length of, 10 width, 14 U UICG, 74, 85, UICG, k-uicg, 220 UISG, 74, 85, UISG, k-uisg,
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