C. Papadimitriou and M. Yannakakis, The clique problem for planar graphs, Inform. Proc. Letters 13 (1981), pp. 131--133.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....useful in some algorithms on graphs. The problems we concentrate on are the subgraph listing problems. We show how this data structure yields linear time algorithms for listing triangles and 4cliques in planar graphs. It has been known before that these two problems can be solved in linear time [20, 6]. However, using our compacted adjacency matrix, both problems become trivial. Let us also point out the connection between our work and the recent paper of Kannan, Naor and Rudich [16] They investigate the problem of labelling the vertices of a graph G in such a way, that given the labels of u ....

....our compacted adjacency matrix can be applied in some algorithms for planar graphs. Consider the two following problems: Given a planar graph G, list all triangles in G. Given a planar graph G, list all 4 cliques in G. There are several algorithm for these problems which use a linear time [2, 15, 20, 6]. However, some of them tend to be rather complicated, especially the algorithm of Papadimitriou and Yannakakis for listing all 4 cliques. In this section we show how we can use a compacted adjacency matrix, together with adjacency lists, for this purpose. We consider only the problem of listing ....

C. Papadimitriou and M. Yannakakis, The clique problem for planar graphs, Inform. Proc. Letters 13 (1981), pp. 131--133.

....useful in some algorithms on graphs. The problems we concentrate on are the subgraph listing problems. We show how this data structure yields linear time algorithms for listing triangles and 4cliques in planar graphs. It has been known before that these two problems can be solved in linear time [20, 6]. However, using our compacted adjacency matrix, both problems become trivial. Let us also point out the connection between our work and the recent paper of Kannan, Naor and Rudich [16] They investigate the problem of labelling the vertices of a graph G in such a way, that given the labels of u ....

....our compacted adjacency matrix can be applied in some algorithms for planar graphs. Consider the two following problems: ffl Given a planar graph G, list all triangles in G. ffl Given a planar graph G, list all 4 cliques in G. There are several algorithm for these problems which use a linear time [2, 15, 20, 6]. However, some of them tend to be rather complicated, especially the algorithm of Papadimitriou and Yannakakis for listing all 4 cliques. In this section we show how we can use a compacted adjacency matrix, together with adjacency lists, for this purpose. We consider only the problem of listing ....

C. Papadimitriou and M. Yannakakis, The clique problem for planar graphs, Inform. Proc. Letters 13 (1981), pp. 131--133.

....subgraph of the K a free graphs if and only if G is (a 1) connected. Our results use a simple Ramsey theoretic lemma that may be of independent interest. 1 Introduction It follows from the sparsity of planar graphs that each such graph contains at most O(n) complete subgraphs K 3 and K 4 [6]. All cliques in a planar graph can be listed by an algorithm with O(n) worst case time complexity [3, 4, 6] Enumeration of subgraphs has a number of uses, including a recent application in testing inscribability [5] These results naturally raise the question of determining which other planar ....

....lemma that may be of independent interest. 1 Introduction It follows from the sparsity of planar graphs that each such graph contains at most O(n) complete subgraphs K 3 and K 4 [6] All cliques in a planar graph can be listed by an algorithm with O(n) worst case time complexity [3, 4, 6]. Enumeration of subgraphs has a number of uses, including a recent application in testing inscribability [5] These results naturally raise the question of determining which other planar graphs occur O(n) times as subgraphs of planar graphs. A necessary condition is that the subgraph G be ....

C. Papadimitriou and M. Yannakakis. The clique problem for planar graphs. Inform. Process Lett. 13 (1981) 131--133.

.... where the maximum clique independent set problem have been studied in the literature can be found in [23] 48] 49] 79] 83] 88] 89] 91] 90] 108] 115] 129] 130] 147] 148] 154] 171] 180] 181] 182] 198] 197] 203] 224] 228] 229] 236] 243] 249] [257], 282] 290] 291] 304] 86] 103] and [325] We should note here that the weighted or unweighted version of the maximum clique problem, the maximum independent set problem, and the minimum vertex cover problem may, with respect to hardness, not be equivalent on graphs with special ....

C. Papadimitriou and M. Yannakakis, The clique problem for planar graphs, Inform. Proc. Lett., Vol. 13: 131--133, 1981.

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