Chiba, N. and Nishizeki, T. Arboricity and subgraph listing algorithms. -- SIAM J. Comput., 14, 1, February 1985, 210-223.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of separating triangles decreases and no such triangles are created anew. The dummy vertices z do not appear in any separating triangle. We perform this operation until all separating triangles are destroyed. The separating triangles can be e#ciently found by the algorithm of Chiba and Nishizeki [4]. Then the new graph G # is fourconnected and triangulated. We now apply the basic technique described in the previous section to G # . The only modification is the handling of the dummy vertices z. Figure 3 gives an example. 2 3 4 5 6 7 12 11 1 8 9 10 7 12 11 1 2 3 4 5 6 8 9 10 12 11 10 9 ....

....at the very least. i th pair e l er G i 1 Figure 5: The recursive definition of the graph with exponential height. We conclude with a note regarding the runtime. Clearly the first part of the construction works in linear time, since we can use the linear time algorithm of Chiba Nishizeki [4] to determine the separating triangles. Then the saving of the third bends by rotating some of the segments might cause a quadratic number of steps. Theorem 3.3 Any plane graph can be mapped on any given point set in the plane and can be drawn with at most three bends per edge in linear time and ....

Chiba, N., and T. Nishizeki, Arboricity and subgraph listing algorithms, SIAM J. Comput. 14 (1985), pp. 210--223.

.... set of k vertices from G which contain a subgraph isomorphic to H, if one exists [10] ffl Enumerative algorithms return a count of how many subgraphs of G are isomorphic to H [6] ffl Enumeration algorithms return all labeled sets of k vertices from G which contain a subgraph isomorphic to H [4]. As we shall see, the existential and constructive problems do not necessarily have the same complexity. Previous work on fixed subgraph isomorphism has concentrated on recognizing specific families of subgraphs, such as small paths [1, 3, 11] cycles [4, 10, 11, 15, 16] and cliques [12] A ....

....which contain a subgraph isomorphic to H [4] As we shall see, the existential and constructive problems do not necessarily have the same complexity. Previous work on fixed subgraph isomorphism has concentrated on recognizing specific families of subgraphs, such as small paths [1, 3, 11] cycles [4, 10, 11, 15, 16], and cliques [12] A practical algorithm for subgraph isomorphism, without analysis, is presented by [19] In Section 2, we consider the complexity of recognizing paths of length k. This leads to a more general algorithm for fixed subgraph isomorphism, presented in Section 3, whose time ....

[Article contains additional citation context not shown here]

N. Chiba and T. Nishizeki. Arboricity and subgraph listing algorithms. SIAM J. Comput, 14(1):210--223, 1985.

....before. This can happen at most twice to every edge. Also the degree of a vertex never decreases and therefore we have a running time of O( added edges q min deg(v) deg(w) O(n) v,w)E where E is the set of edges in the final graph and the last equality is due to Chiba and Nishizeki [2]. Now, we prove the correctness of the algorithm: Lemma 3.4 The graph stays simple. Proof: Notice that the edge (Ul, up) does not exist. Assume it did. Then v must have degree 2, otherwise we had a separating triangle. But Ul has degree at least 3, which contradicts the choice of v. So we ....

....a biconnected planar graph G, it is NP complete to decide whether G can be embedded such that the number of separating triangles of G is at most k. Proof: The problem is in NP. Given a planar embedding of G we can count the number of separating triangles in the embedding in polynomial time (see [2]) Let G be an arbitrary triangular planar graph. For every edge (a, b) C G, we add a vertex x with edges to a and to b. Let G be the resulting graph. Clearly G t is biconnected and planar. Let F and F be the two faces incident to (a, b) in G. If we place x inside F then F is a separating ....

Chiba, N., and T. Nishizeki, Arboricity and subgraph listing algorithms,

....all edges we find that at most X (v i ;v j )2E 2 minfffi v i Gamma 1; ffi v j Gamma 1gd = O(ffied ) revisions will be performed, where ffi is the maximum degree. This should be compared to the O(n ) revisions performed by the classical PC algorithm. Using the results presented in [ Chiba and Nishizeki, 1985 ] one may use the arboricity ff of G instead of ffi, resulting in a number of revisions O(ffed ) The same reference also presents upper bounds on ff both for general graphs and for specific types of graphs. For the experiments below we report the number of revisions since this measure is ....

N. Chiba and T. Nishizeki. Arboricity and subgraph listing algorithms. SIAM Journal on Computing, 14, 1985.

.... 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 methods also generalize to problems of nding K 2;2 and other complete bipartite subgraphs [12, 17] Richards [42] gives O(n log n) algorithms for nding C 5 and C 6 subgraphs in planar graphs, and leaves open the question for larger cycle lengths; Alon et al. 1] gave O(n log n) ....

.... 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 methods also generalize to problems of nding K 2;2 and other complete bipartite subgraphs [12, 17]. Richards [42] gives O(n log n) algorithms for nding C 5 and C 6 subgraphs in planar graphs, and leaves open the question for larger cycle lengths; Alon et al. 1] gave O(n log n) deterministic and O(n) randomized algorithms for larger cycles. In [16] we showed how to list all cycles of a ....

N. Chiba and T. Nishizeki. Arboricity and subgraph listing algorithms. SIAM J. Computing 14:210-223, 1985.

....only if its oriented dual H (which is computed in linear time at line 8: has no circuit. This test (line 9: can be done in linear time using a topological sort. # 4 Enumerations of the triangles of a planar graph Linear time algorithms enumerating the triangles of planar graphs may be found in [1] (using tree decompositions) or in [2] using indegree bounded orientations) The algorithm we present here has been optimized using Schnyder s decompositions, the definition of which we shall recall here: Definition 4.1 (Schnyder, 14] Let G be a maximal planar graph and r 1 , r 2 , r 3 ....

....is not a face of A(G) as it does not correspond to an edge of G. # Remark 5. 2 There will be no linear time algorithm to enumerate the C 4 of 3 connected planar graphs, as this number may be quadratic (any double wheel will do) although it is possible to implicitly enumerate them in linear time [1][4] Lemma 5.3 Let G be a 2 connected planar graph with at least 4 vertices and let A(G) its angle graph, oriented in such a way that each of its vertices have indegree 2, except the vertices of the external faces which have indegree 1. Then, the graph G is 3 connected if and only if A(G) has ....

N. Chiba and T. Nishizeki, Arboricity and subgraph listing algorithms, SIAM J. Computing vol. 14 (1985), 210--223.

....this result with Theorem 3.2 the result follows. 2 4 Recognizing square edge graphs In this section we present an algorithm of complexity O(a(G)m) which recognizes square edge graphs, and find a square edge sequence, if one exists. The algorithm depends on the work of Chiba and Nishizeki [3]. The first part of our algorithm is basically analogous to Algorithm C4 from [3] It finds all the quadrangles of a given graph and prepare some data structure for the second part. For each vertex v of a graph it finds all the quadrangles containing v. Let w be a vertex with dG (v; w) 2. Then, ....

....graphs In this section we present an algorithm of complexity O(a(G)m) which recognizes square edge graphs, and find a square edge sequence, if one exists. The algorithm depends on the work of Chiba and Nishizeki [3] The first part of our algorithm is basically analogous to Algorithm C4 from [3]. It finds all the quadrangles of a given graph and prepare some data structure for the second part. For each vertex v of a graph it finds all the quadrangles containing v. Let w be a vertex with dG (v; w) 2. Then, as square edge graphs contain no K 2;3 as a subgraph, there are at most two ....

[Article contains additional citation context not shown here]

Chiba, N., Nishizeki T.: Arboricity and subgraph listing algorithms. SIAM J. Comput. 14, 210--223 (1985)

....In Section 3 we provide a new proof of the fact that all triangles of a graph can be found in O(m 3=2 ) time. It is similar to the one of Itai and Rodeh and is included for the sake of completeness. Another algorithm which lists all the triangles of a given graph is due to Chiba and Nishizeki [5]. For a graph G its time complexity is O(a(G)m) where a(G) denotes the arboricity of G. They also show that a(G) O(m 1=2 ) Thus, the algorithm of Chiba and Nishizeki is in the worst case still of complexity O(m 3=2 ) We continue this paper as follows. We first recall several notions ....

....H i and perform certain checks, the complexity of these operations being determined by the complexity finding all triangles. Now, the number of triangles in planar graphs can be found in linear time. This is well known and can be shown directly by a modification of Theorem 5. For a reference see [5]. Now the proof is completed by the observation that the total number of edges in the H i is at most 3n, where n is the number of vertices of G. 2 ....

N. Chiba and T. Nishizeki, Arboricity and subgraph listing algorithms, SIAM J. Comput., 14 (1985), pp. 210--223.

....algorithm. Suppose now that G is accepted. Then, by Lemma 7, Theta = ffi which implies that for any edge uv, U uv = U uv . Since the U uv s are checked for being isometric trees, G is a prism free almost median graph. It remains to determine the algorithm s complexity. With the algorithm of [6] we first construct all quadrangles of G. The complexity of the algorithm is O(m a(G) It considers every edge uv and all edges incident with an endpoint of uv of degree minfdG (u) dG (v)g. Since X uv2E(G) minfdG (u) dG (v)g 2a(G)m; and a(G) log n for partial cubes, we can stop the ....

N. Chiba and T. Nishizeki, Arboricity and subgraph listing algorithms, SIAM J. Comput. 14 (1985) 210--223.

....and Rodeh [IR78] showed that a triangle (a C 3 ) in a graph G = V; E) that contains one can be found in O(V ) or O(E 3=2 ) time. We improve their second result and show that the same can be done, in directed or undirected graphs, in O(E 2 1 ) O(E 1:41 ) time. Chiba and Nishizeki [CN85] showed that triangles (C 3 s) and quadrilaterals (C 4 s) in graphs that contain them can be found in O(E Deltad(G) time. As d(G) O(E 1=2 ) for any graph G, this extends the result of Itai and Rodeh. We extend the result of Chiba and Nishizeki and show that C 4k Gamma1 s and C 4k s can ....

....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 of Richards and show that C 5 s in planar graphs can be found in O(V ) time. In a ....

N. Chiba and L. Nishizeki. Arboricity and subgraph listing algorithms. SIAM Journal on Computing, 14:210--223, 1985.

.... E) 106] We can easily improve the dynamic programming algorithm from O(n 3 ) to O(En) but we can even do a little better. The time bound depends on the number 18 of triangles in the polygon; below we show that few edges imply few triangles, a fact previously observed by Chiba and Nishizeki [52]. Lemma 7. A graph with E edges has at most O(E 3=2 ) triangles. Proof: Divide the vertices into two classes: heavy vertices with degree at least p E, and light vertices with smaller degree. If b is light we enumerate the triangles containing b by examining each pair of edges ab, bc, and ....

N. Chiba and T. Nishizeki. Arboricity and subgraph listing algorithms. SIAM J. Computing 14 (1985) 210--223.

....case, the time needed to produce the subproblems from G dominates the time needed to recover a solution from the subproblem solutions, so we ignore the latter. By Corollary 2. 5 (with k = 4) G has m = O(n) edges and arboricity ff(G) O(1) so we can list its O(n) maximal cliques in linear time [7]. From the listed MC 4 s, we can precompute the sets E [a; b] for all unmarked edges fa; bg, again in linear time. We claim that testing the existence of a separating triangle takes O(n 2 ) time. Since G has O(n) maximal cliques and no 7 clique, it has O(n) 3 cliques and these can be found in ....

....to test whether some (ordered) list of the vertices in C is a separating triangle. So, the claim holds. A similar analysis applies for finding a 3 cut (by Lemma 4.3(1) a separating edge, or a separating triple. In order to detect separating quadruples, we use an algorithm of Chiba and Nishizeki [7] which implicitly lists all 4 cycles of G in O(m Delta ff(G) O(n) time. The algorithm produces a list of triples (u i ; v i ; S i ) with the following properties: 1. u i and v i are non adjacent vertices of G. 2. S i is a set of vertices adjacent to both u i and v i . 43 3. Every induced ....

N. Chiba and T. Nishizeki. Arboricity and subgraph listing algorithms. SIAM J. Computing, 14(1):210--223, 1985. 45

....the cycles in C are boundaries of faces. Two corollaries of this result are worth mentioning. First, it can be decided in linear time whether a graph G can be embedded such that all triangles are faces: Chiba and Nishiseki showed how to enumerate all triangles of a planar graph in linear time [5], and then we apply the above algorithm with C as the set of all triangles. This result was shown before [2] but the approach presented here avoids splitting the graph into its triconnected components, and hence is simpler. Knowledge of such an embedding enables us to make a planar graph ....

N. Chiba and T. Nishizeki. Arboricity and subgraph listing algorithms. SIAM J. Computing, 14(1):210--223, 1985.

....listing of all triangles. Related subgraph isomorphism problems also occur in a wide variety of practical applications [2, 5, 12, 9, 13, 14, 19] For planar graphs, or more generally for graphs of bounded arboricity, the problem of listing clique subgraphs is well understood. Chiba and Nishizeki [6] show that there can be at most O(n) cliques of a given size in such graphs, and they further describe linear time algorithms for listing these cliques. An alternative linear time algorithm and its parallelization is presented in [7] Enumeration of incomplete subgraphs is less well understood, ....

.... for which it is known how to list the occurrences of G in linear time are cliques (as noted above) and wheels [10] Chiba and Nishizeki studied C 4 subgraphs in graphs of bounded arboricity; there can be #(n 2 ) such 4 cycles but an implicit representation of them can be found in linear time [6]. In this paper we again consider listing subgraphs of bounded arboricity graphs. The arboricity a(G) of a graph G is the minimum number of forests into which the edges of G can be partitioned [16] Every planar graph has arboricity at most three [16] many other classes of graphs enjoy a bounded ....

N. Chiba and T. Nishizeki. Arboricity and subgraph listing algorithms. SIAM J. Computing, 14:210--223, 1985.

....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 ....

....linear time (matching our bounds on the occurrences of G) Such algorithms were already known for enumerating copies of the complete graphs K 3 and K 4 as subgraphs of planar graphs. More generally, in any family of graphs with bounded arboricity, all clique subgraphs can be listed in linear time [3, 4]. The family of K 1,b free graphs is also relatively easy. Since these graphs have maximum vertex degree b 1, there are fewer than b k vertices within distance k of any given vertex. The occurrences of any connected subgraph G can therefore be enumerated in time O(b k n) O(n) The first ....

N. Chiba and T. Nishizeki. Arboricity and subgraph listing algorithms. SIAM J. Comput. 14 (1985) 210--223.

.... 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 also generalize to problems of finding K 2,2 and other complete bipartite subgraphs [11, 16] In [15] we showed how to list all cycles of a given fixed length in outerplanar graphs, in linear time (see also [29, 30, 31, 39] for similar variants of outerplanar subgraph ....

.... 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 also generalize to problems of finding K 2,2 and other complete bipartite subgraphs [11, 16]. In [15] we showed how to list all cycles of a given fixed length in outerplanar graphs, in linear time (see also [29, 30, 31, 39] for similar variants of outerplanar subgraph isomorphism) We used our outerplanar cycle result to find any wheel of a given fixed size in planar graphs, in linear ....

N. Chiba and T. Nishizeki. Arboricity and subgraph listing algorithms. SIAM J. Computing, 14:210--223, 1985.

....of separation triangles decreases and no such triangles are created anew. The dummy vertices z do not appear in any separating triangle. We perform this operation until all separating triangles are destroyed. The separating triangles can be eciently found by the algorithm of Chiba and Nishizeki [4]. Then the new graph G 0 is four connected. We now apply the technique described above to G 0 . The only modi cation is the handling of the dummy vertices z. Figure 3 gives an example. Let C 0 be the external hamiltonian cycle as found by the algorithm of Chiba Nishizeki. Clearly, C 0 ....

....at the very least. i th pair e l er G i 1 Fig. 5. The recursive de nition of the graph with exponential height. We conclude with a note regarding the runtime. Clearly the rst part of the construction works in linear time, since we also can use the linear time algorithm of Chiba Nishizeki [4] to determine the separating triangles as well. Then the saving of the third bends by rotating some of the segments might cause a quadratic number of steps. Theorem 4. Any plane graph can be mapped on any given point set in the plane and can be drawn with at most three bends per edge in linear ....

Chiba, N., and T. Nishizeki, Arboricity and subgraph listing algorithms, SIAM J. Comput. 14 (1985), pp. 210-223.

.... 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 methods also generalize to problems of nding K 2;2 and other complete bipartite subgraphs [12, 17] Richards [42] gives O(n log n) algorithms for nding C 5 and C 6 subgraphs in planar graphs, and leaves open the question for larger cycle lengths; Alon et al. 1] gave O(n log n) ....

.... 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 methods also generalize to problems of nding K 2;2 and other complete bipartite subgraphs [12, 17]. Richards [42] gives O(n log n) algorithms for nding C 5 and C 6 subgraphs in planar graphs, and leaves open the question for larger cycle lengths; Alon et al. 1] gave O(n log n) deterministic and O(n) randomized algorithms for larger cycles. In [16] we D. Eppstein, Planar Subgraph ....

N. Chiba and T. Nishizeki. Arboricity and subgraph listing algorithms. SIAM J. Computing 14:210-223, 1985.

....the number of edges in the graph. A very basic problem in theoretical computer science is the problem of finding a triangle in a graph. In 1978 Itai and Rodeh presented two solutions for this problem. The first algorithm has a running time of O(n ff ) The second algorithm needs O(e 3=2 ) In [2] this second result was refined to O(ea(G) where a(G) is the arboricity of the graph. Since a(G) O( p e) in a connected graph (see [2] this extends the result of Itai and Rodeh. These were for almost ten years the best known algorithms. A drastic improvement was made recently by Alon et. ....

....Itai and Rodeh presented two solutions for this problem. The first algorithm has a running time of O(n ff ) The second algorithm needs O(e 3=2 ) In [2] this second result was refined to O(ea(G) where a(G) is the arboricity of the graph. Since a(G) O( p e) in a connected graph (see [2]) this extends the result of Itai and Rodeh. These were for almost ten years the best known algorithms. A drastic improvement was made recently by Alon et al. In [1] they showed the following surprisingly elegant and easy result. Deciding whether a directed or an undirected graph G = V; E) ....

Chiba, N. and T. Nishizeki, Arboricity and subgraph listing algorithms, SIAM J. Comput., 14, (1985), pp. 210--223.

....cycle containing f 0 and not e, contradicting the invariant. The case in which C contains f 0 and e, but not f , is similar. It follows that S is a cactus. For each two edges in F we get a triangle in S, for a total of p triangles. This completes the proof. As described by Chiba and Nishizeki [CN85], we can explicitly list all the triangles in a graph G with m edges in time O(m 3=2 ) So jE 0 j is O(m 3=2 ) Gabow and Stallmann [GS85] describe an algorithm for GMP, which runs in time O(m 0 n 0 log 6 n 0 ) where m 0 and n 0 are the number of edges and vertices, ....

N. Chiba and T. Nishizeki, "Arboricity and Subgraph Listing Algorithms," SIAM Journal of Computing, 14:210-223, 1985.

....of E(G) n E(P ) to P , modifying the embedding of P , so that its genus increases by at most one. Step 4: Output P and the embedding of P of genus at most d. Note that the algorithm can be implemented in polynomial time, since one can explicitly list all the triangles of G in time jE(G)j 3 2 [CN85]. Also steps 2 and 3 iterate at most jE(G)j times. Theorem 11. The approximation ratio of algorithm B is 1 4 . We omit the proof of this theorem by lack of space. 4 Open Problems To our knowledge, nothing is known about a polynomial time algorithm with constant approximation ratio for finding ....

N. Chiba and T. Nishizeki, "Arboricity and Subgraph Listing Algorithms", SIAM Journal of Computing, 14:210-223, 1985.

....forest F in G 0 with 2p edges if and only if there is a triangular cactus S in G with p triangles. Moreover, S can be obtained from F (and vice versa) in time O(n) For lack of space, we omit the proof of this lemma, which is used implicitly in [LP86] As described by Chiba and Nishizeki [CN85], we can explicitly list all the triangles in a graph G with m edges in time O(m 3=2 ) So jE 0 j is O(m 3=2 ) Gabow and Stallmann [GS85] describe an algorithm for GMP, which runs in time O(m 0 n 0 log 6 n 0 ) where m 0 and n 0 are the number of edges and vertices, ....

N. Chiba and T. Nishizeki, "Arboricity and Subgraph Listing Algorithms," SIAM Journal of Computing, 14:210-223, 1985.

....O(V ) expected time or O(V log V ) worst case time and even in O(V ) worst case time if k 5. This improves and extends an 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 ....

....by three colors and a two colored graph is obtained. A well colored C 2 in such a graph is just a pair of parallel edges and such a pair, if one exists, can be easily found in O(V ) time. Alternatively, we can stop the recursion when k = 3 and use an existing O(E Delta d(G) time algorithm (see [CN85] for finding triangles (C 3 s) in a general graph G = V; E) Note that any triangle in a three colored graph is well colored and that O(E Delta d(G) is O(V ) in our case. 2 The algorithm just described can again be derandomized. Theorem 5.3 Let C be a non trivial minor closed family of ....

N. Chiba and L. Nishizeki. Arboricity and subgraph listing algorithms. SIAM Journal on Computing, 14:210--223, 1985.

....0 and not e; e 0 ; e 00 , contradicting the invariant. The case in which C contains e 0 and e 00 but not e is similar. It follows that S is a cactus. For each two edges in F we get a triangle in S, for a total of p triangles. This completes the proof. As described by Chiba and Nishizeki [CN85], we can explicitly list all the triangles in a graph G with m edges in time O(m 3=2 ) So jE 0 j is O(m 3=2 ) Gabow and Stallmann [GS85] describe an algorithm for GMP, which runs in time O(m 0 n 0 log 6 n 0 ) where m 0 and n 0 are the number of edges and vertices, ....

N. Chiba and T. Nishizeki, "Arboricity and Subgraph Listing Algorithms," SIAM Journal of Computing, 14:210-223, 1985.

....algorithm of Kanevsky et al. which builds a 4 block tree of a general graph in O(n Delta ff(m; n) m) time [9] In our case a separating triplet is a separating triangle, which forms the basis for the algorithm. For determining the separating triangles, we use the algorithm of Chiba Nishizeki [2] for determining triangles in a graph. In [17] Richards describes another linear time algorithm. Chiba Nishizeki first sort the vertices in v 1 ; v n in such a way that deg(v 1 ) deg(v 2 ) deg(v n ) Observing that each triangle containing vertex v i corresponds to an edge ....

....in Adj(v) which can be tested in O(1) time by maintaining crosspointers. To compute the time complexity of this algorithm, Chiba Nishizeki use the arboricity of G, defined as the minimum number of edge disjoint forests into which G can be decomposed, and denoted by a(G) 7] Lemma 4. 1 ([2]) P (u;v)2E minfdeg(u) deg(v)g 2 Delta a(G) Delta m. Using this lemma Chiba Nishizeki show that the time complexity of the algorithm is O(a(G) Delta m) If G is planar then a(G) 3 ( 7] so the algorithm runs in O(n) time in case G is planar. A B C D E F G L M G H I J K L E ....

Chiba, N., and T. Nishizeki, Arboricity and subgraph listing algorithms, SIAM J. Comput. 14 (1985), pp. 210--223.

....Itai and Rodeh [8] showed that a triangle (a C 3 ) in a graph G = V; E) that contains one can be found in O(V ) or O(E 3=2 ) time. We improve their second result and show that the same can be done, in directed or undirected graphs, in O(E 2 1 ) O(E 1:41 ) time. Chiba and Nishizeki [6] showed that triangles (C 3 s) and quadrilaterals (C 4 s) in graphs that contain them can be found in O(E Delta d(G) time. As d(G) O(E 1=2 ) for any graph G, this extends the result of Itai and Rodeh. We extend the result of Chiba and Nishizeki and show that C 4k Gamma1 s and C 4k s can ....

....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 Richards and show that C 5 s in planar graphs can be found in O(V ) time. In a previous ....

N. Chiba and L. Nishizeki. Arboricity and subgraph listing algorithms. SIAM Journal on Computing, 14:210--223, 1985.

....only if its oriented dual H (which is computed in linear time at line 8: has no circuit. This test (line 9: can be done in linear time using a topological sort. 3. Enumerations of the triangles of a planar graph Linear time algorithms enumerating the triangles of planar graphs may be found in [1] (using tree decompositions) or in [2] using indegree bounded orientations) The algorithm we present here has been optimized using Schnyder s decompositions, the de nition of which we shall recall here: 4 H. DE FRAYSSEIX AND P. OSSONA DE MENDEZ De nition 3.1. Let G be a maximal planar graph ....

N. Chiba and T. Nishizeki, Arboricity and subgraph listing algorithms, SIAM J. Computing 14 (1985), 210-223.

....studied in section 6. A number of other clique generating algorithms were brought to our notice after we had chosen the above algorithm. The algorithm proposed in [8] was shown to have superior performance than the Bierstone algorithm. Some other newer algorithms for this problem are presented in [10, 30]. We plan to incorporate these for the clique generation step of our algorithm, minimizing any overhead due to this step. 1:for i = N ; i = 1; i Gamma Gamma do 2: i] CliqList = 3: for all x 2 [i] CoveringSet do 4: for all cliq 2 [x] CliqList do 5: M = cliq [i] 6: if M 6= then 7: ....

....the clique based schemes will suffer. This is borne out in the graphs for T20.I2.D100K with decreasing support, and in figure 11 b) as the transaction size increases for a fixed support value. We expect to reduce the overhead of the clique generation by implementing the algorithms proposed in [8, 10, 30], which were shown to be superior to the Bierstone algorithm [19] a modification of which is used in the current implementation. 6.2 Join and Memory Statistics 1000 10000 100000 1e 06 T10.I2 T10.I4 T20.I4 T20.I6 T20.I2 Number of Intersections Databases Min Support: 0.25 Partition 10 Eclat ....

N. Chiba and T. Nishizeki. Arboricity and subgraph listing algorithms. In SIAM J. Computing, 14(1):210-223, Feb. 1985.

....related to fixed vertices of G. Then, cliques were generated from these partial cliques. Their computational results suggested their proposed algorithm was as efficient as that in [73] for general graphs, but more efficient on sparse graphs. In 1980 s, other proposed algorithms include those in [216, 215, 87, 310, 191]. Loukakis and Tsouros [216] proposed a depth first enumerative algorithm that generated all maximal independent sets lexicographically. They compared their algorithm with the algorithms of [73] and [311] Their computational results on graphs of up to 220 vertices suggested the superior ....

....(for large graphs) In 1988, Johnson et al. 191] proposed an algorithm that enumerated all maximal independent sets in lexicographic order. The algorithm has an O(n 3 ) delay between the generation of two subsequent independent sets; cf. the complexity result Theorem 3.1. Chiba and Nishizeki s [87] algorithm lists all cliques with time complexity of O(a(G)m ) where a(G) is the arboricity of graph G. This is an improvement over the time complexity in [311] Finally, Tomita et al. 310] proposed a modified Bron and Kerbosch [73] algorithm and claimed its time complexity to be O(3 n=3 ) ....

N. Chiba and T. Nishizeki, Arboricity and subgraph listing algorithms, SIAM J. Comput., Vol. 14: 210--223, 1985.

No context found.

Chiba, N. and Nishizeki, T. Arboricity and subgraph listing algorithms. -- SIAM J. Comput., 14, 1, February 1985, 210-223.

No context found.

N. Chiba and L. Nishizeki. Arboricity and subgraph listing algorithms. SIAM Journal on Computing, 14:210{ 223, 1985.

No context found.

N. Chiba and L. Nishizeki. Arboricity and subgraph listing algorithms. SIAM Journal on Computing, 14:210-223, 1985.

No context found.

N. Chiba and T. Nishizeki, "Arboricity and Subgraph Listing Algorithms," SIAM Journal of Computing, 14:210--223, 1985.

No context found.

N. Chiba and T. Nishizeki, "Arboricity and Subgraph Listing Algorithms," SIAM Journal on Computing 14 (1985), 210--223.

No context found.

N. Chiba and T.Nishizeki, Arboricity and subgraph listing algorithms, SIAM J. Comput. 14 (1985), pp. 210--223.

No context found.

N. Chiba and T.Nishizeki, Arboricity and subgraph listing algorithms, SIAM J. Comput. 14 (1985), pp. 210--223.

No context found.

N. Chiba and T. Nishizeki, "Arboricity and Subgraph Listing Algorithms," SIAM Journal on Computing 14 (1985), 210-223.

No context found.

Chiba, N. and T. Nishizeki, Arboricity and subgraph listing algorithms, SIAM J. Comput. 14, (1985), pp. 210--223.

No context found.

N. Chiba and T. Nishizeki, "Arboricity and Subgraph Listing Algorithms," SIAM Journal on Computing 14 (1985), 210-223.

No context found.

N. Chiba and T. Nishizeki, \Arboricity and Subgraph Listing Algorithms," SIAM Journal on Computing 14 (1985), 210-223.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC