C. Papadimitriou. Ecient search for rationals. Information Processing Letters, 8:1-4, 1979.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....show that C 4k 1 s can be found in O(E 2 Gamma 1 k Delta d(G) 1 1 k ) time. This gives, in particular, an O(E Delta d(G) 2 ) algorithm for finding pentagons (C 5 s) Our results apply to both directed and undirected graphs. Itai and Rodeh [IR78] and also Papadimitriou and Yannakakis [PY81] showed that C 3 s in planar graphs can be found in O(V ) time. Chiba and Nishizeki [CN85] showed that C 3 s as well as C 4 s in planar graphs can be found in O(V ) time. Richards [Ric86] showed that C 5 s and C 6 s in planar graphs can be found in O(V log V ) time. We improve upon the result ....

C.H. Papadimitriou and M. Yannakakis. The clique problem for planar graphs. Information Processing Letters, 13:131--133, 1981.

....LP by picking any cost vector c, say c = 0, and checking whether the resulting linear program is feasible. Remarks: The original reduction in [DLR79] is from HORN, Problem A.6.2, to LP. In [DR80] LP and LI are shown to be log space equivalent by reducing LP to LI using rational binary search [Pap78, Rei78] to find the value of the maximum and an x that Part II: P Complete Problems ffl 63 yields it. However, it is not clear how to perform this reduction in NC 1 . Since LP and LI are complete via NC 1 reductions though, there must be a NC 1 reduction between the two problems. Although we know ....

C. H. Papadimitriou. Efficient search for rationals. Information Processing Letters, 8(1):1--4, 1978.

....we achieve. Several papers have studied planar subgraph isomorphism with restricted patterns. It has long been known that if the pattern H is either K 3 or K 4 , then there can be at most O(n) instances of H as a subgraph of a planar graph G, and that these instances can be listed in linear time [6, 22, 32], a fact which has been used in algorithms to test connectivity [27] approximate maximum independent sets [6] and test inscribability [13] Linear time and instance bounds for K 3 and K 4 can be shown to follow solely from the sparsity properties of planar graphs [11, 12] and similar methods ....

....the maximum clique by finding a set of h vertices inducing as many edges as possible. The connected h clustering problem adds the restriction that the induced subgraph be connected. Keil and Brecht [23] study these problems, and show that even though cliques are easy to find in planar graphs [32], the connected h clustering problem is NP complete for planar graphs. See [25] for approximate h clustering algorithms in general graphs. One method for exact solution to the h clustering problem is simply to test subgraph isomorphism for all possible planar graphs on h vertices. Corollary 2. ....

C. H. Papadimitriou and M. Yannakakis. The clique problem for planar graphs. Information Processing Letters, 13:131--133, 1981.

....LP by picking any cost vector c, say c = 0, and checking whether the resulting linear program is feasible. Remarks: The original reduction in [DLR79] is from HORN, Problem A.6.2, to LP. In [DR80] LP and LI are shown to be log space equivalent by reducing LP to LI using rational binary search [Pap78, Rei78] to find the value of the maximum and an x that Part II: P Complete Problems ffl 63 yields it. However, it is not clear how to perform this reduction in NC 1 . Since LP and LI are complete via NC 1 reductions though, there must be a NC 1 reduction between the two problems. Although we know ....

C. H. Papadimitriou. Efficient search for rationals. Information Processing Letters, 8(1):1--4, 1978.

.... O(V log V ) worst case bound, for k = 5; 6, obtained by Richards [Ric86] using planar separators and an O(V ) worst case bound, for k = 3; 4, obtained by Chiba and Nishizeki [CN85] Algorithms for finding triangles in planar graphs in O(V ) time were also obtained by Papadimitriou and Yannakakis [PY81] and Itai and Rodeh [IR78] Our initial goal was to obtain efficient algorithms for finding simple paths and cycles in graphs. The algorithms we developed using the color coding method turned out however to have a much wider range of applicability. The linear time (i.e. 2 O(k) Delta E for ....

C.H. Papadimitriou and M. Yannakakis. The clique problem for planar graphs. Information Processing Letters, 13:131--133, 1981.

....show that C 4k 1 s can be found in O(E 2 Gamma 1 k Delta d(G) 1 1 k ) time. This gives, in particular, an O(E Delta d(G) 2 ) algorithm for finding pentagons (C 5 s) Our results apply to both directed and undirected graphs. Itai and Rodeh [8] and also Papadimitriou and Yannakakis [12] showed that C 3 s in planar graphs can be found in O(V ) time. Chiba and Nishizeki [6] showed that C 3 s as well as C 4 s in planar graphs can be found in O(V ) time. Richards [13] showed that C 5 s and C 6 s in planar graphs can be found in O(V log V ) time. We improve upon the result of ....

C.H. Papadimitriou and M. Yannakakis. The clique problem for planar graphs. Information Processing Letters, 13:131--133, 1981.

....we achieve. Several papers have studied planar subgraph isomorphism with restricted patterns. It has long been known that if the pattern H is either K 3 or K 4 , then there can be at most O(n) instances of H as a subgraph of a planar graph G, and that these instances can be listed in linear time [7, 28, 40], a fact which has been used in algorithms to test connectivity [35] to approximate maximum independent sets [7] and to test inscribability [14] Linear time and instance bounds for K 3 and K 4 can be shown to follow solely from the sparsity properties of planar graphs [12, 13] and similar ....

....the maximum clique by nding a set of h vertices inducing as many edges as possible. The connected h clustering problem adds the restriction that the induced subgraph be connected. Keil and Brecht [30] study these problems, and show that even though cliques are easy to nd in planar graphs [40], the connected h clustering problem is NPcomplete for planar graphs. See [32] for approximate h clustering algorithms in general graphs. Figure 5: An embedded planar graph G, the vertex connectivity substitute (with edges drawn as heavy curves) and the edge connectivity substitute . ....

C. H. Papadimitriou and M. Yannakakis. The clique problem for planar graphs. Information Processing Letters 13:131-133, 1981.

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C. Papadimitriou. Ecient search for rationals. Information Processing Letters, 8:1-4, 1979.

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C.H. Papadimitriou and M. Yannakakis. The clique problem for planar graphs. Information Processing Letters, 13:131-133, 1981.

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