J. D. Horton. A polynomial-time algorithm to find a shortest cycle basis of a graph. SIAM Journal of Computing, 16:359--366, 1987.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of practical applications. They play a crucial role in chemical ring perception [6] structural flexibility analysis [15] electrical networks [5] and error propagation in Address for correspondence chemical reaction networks [8] Brief surveys and extensive references can be found e.g. in [12, 13]. In many cases the network graphs of interest are intrinsically directed. It is natural therefore to ask for a description of the cycle structure in terms of circuits, i.e. cycles that are following the directions of the arcs. This problem is of particular interest for the analysis of the ....

....of circuits in a digraph G(V, A) may be very large, however. The straightforward application of the greedy algorithm or of Algorithm 1 to the set of all circuits will therefore not be feasible in most cases. In the case of undirected graphs one can drastically reduce the initial set of cycles [2, 13, 23]. In the following section we consider similar constructions for circuits in digraphs. The main di#erence is that in the undirected graph one can work over GF (2) and explicitly use the vector addition of cycles. Here we have the additional problem that the sum of circuits is in general not a ....

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J. D. Horton. A polynomial-time algorithm to find the shortest cycle basis of a graph. SIAM J. Comput., 16:359--366, 1987.

.... bases of the cycle space of a graph (MCBs) have a variety of applications in science and engineering, for example, in structural flexibility analysis [15] electrical networks [5] and in chemical structure storage and retrieval systems [6] Brief surveys and extensive references can be found in [13, 12]. In general, minimum cycle bases are not very well behaved under graph operations. Neither the total length #(G) nor the length of the longest cycle #(G) in a MCB of G are minor monotone, see Fig. 1 for a counterexample. Hence, there does not seem to be a general way of extending MCBs of a ....

....= C#B C . A minimum cycle basis (MCB) is a cycle basis with minimum length. Since the cycle space is a matroid in which an element has weight the greedy algorithm can be used to extract a MCB, see e.g. 20] The cycles in are edge connected, chordless, and isometric, see e.g. [13]. An MCB can be computed in polynomial time [13, 1] A cycle is relevant if it is contained in some MCB [18] Proposition 1. 19] A cycle C is relevant if and only if it cannot be written as a of shorter cycles. 2.2. Products. Given two non empty graphs G = V G , EG ) and H = V H , EH ) the ....

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J. D. Horton. A polynomial-time algorithm to find the shortest cycle basis of a graph. SIAM J. Comput., 16:359--366, 1987.

.... correspond to productive pathways [2, 4, 7] Hartvigsen [5] introduced the U space U(G) as the union of U # and the cycle space C(G) He gives an algorithm for computing a minimum length basis of U(G) a minimum U basis for short, in polynomial time that extends a previous algorithm by Horton [6] for minimum length bases of the C(G) More recently, Vismara [8] showed how to compute the set of relevant cycles, i.e. the union of all minimum length bases of C(G) using a method that is based on Horton s algorithm. It is the purpose of this note is to extend Vismara s approach to the ....

....the set RV of V relevant subgraphs is the union of all minimum length bases of V. Lemma 5. A V is relevant if and only if A cannot be written as the of strictly shorter elements of V. Proof. The proof of Vismara s [8] Lemma 1 is valid for arbitrary vector spaces of subgraphs. # Horton s [6] mimimal cycle basis algorithm is based on an easy to check necessary condition for relevance: A cycle is edge short if it contains an edge e = y and a vertex z such that C P xz P yz where P xz and P yz are shortest paths . Hartvigsen [5] generalized this notion to paths: A ....

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J. D. Horton. A polynomial-time algorithm to find the shortest cycle basis of a graph. SIAM J. Comput., 16:359--366, 1987.

....lie) it can be seen that the amount of computation involved in our mesh analysis algorithm is proportional to the length of the cycle set. Unfortunately, Deo et al. [4] proved the problem of finding a spanning tree whose fundamental set of cycles has total shortest length to be NP complete. Horton [11] described a polynomial time algorithm for finding the shortest cycle basis of a graph with m number of edges and n vertices in operations, with a formidable worst case of . The analysis in [11] was restricted to 2 connected graphs without loops or multiple edges. Since then, Thomassen [24] went ....

....a spanning tree whose fundamental set of cycles has total shortest length to be NP complete. Horton [11] described a polynomial time algorithm for finding the shortest cycle basis of a graph with m number of edges and n vertices in operations, with a formidable worst case of . The analysis in [11] was restricted to 2 connected graphs without loops or multiple edges. Since then, Thomassen [24] went on to prove the problem of finding a cycle cover of smallest total length for an arbitrary graph to be NP hard. The solution to least norm perturbation is, however, not dependent of choice of ....

Horton J.D., A polynomial-time algorithm to find the shortest cycle basis of a graph, SIAM J.Computing, Vol.16, No,2, April 1987.

....is neither part of another 2 cycle nor part of a cycle in B. We have B # D = #(G # ) D = E V c(G # ) R = A V c(G # ) #( # G) hence B # D is a basis of the circuit space. Now we can use the well known fact that the circuits form a matroid (see e.g. (Horton, 1987; Hartvigsen and Mardon, 1993) to obtain minimal length cycle bases by means of the greedy algorithm from the set Z of all circuits. The crucial observation is that Z consists of the double edges, i.e. 2 cycles, and of circuits that are obtained from the cycles of G by omitting a particular ....

....of R(G) in terms of number and length distribution of cycles is an important characteristic of a graph. The numerical studies below 8 P.M. Gleiss, P.F. Stadler, A. Wagner, D.A. Fell make use of Vismara s (Vismara, 1997) algorithm for computing R(G) which is based on Horton s MCB algorithm (Horton, 1987). 3. Flux Analysis in Chemical Networks Let us now return to chemical networks. Because it is germane to their functional analysis, we first point out a nexus between graph representations of metabolic network, and metabolic flux analysis (MFA) the most generic framework to analyze the ....

Horton, J. D. A polynomial-time algorithm to find the shortest cycle basis of a graph. SIAM J. Comput. 16, 359--366 (1987).

....Delete k edges, one for each cycle, in a combinatorial manner. Apply the dynamic programming between the resulting tree and the pattern tree to find the maximum number of pattern trees, disjointly packed in the structure. The first step, the detection of cycles, is performed by Horton s algorithm [9]. Its original purpose is to find a minimum cycle basis, and the final Gaussian elimination is required to remove such cycles that can be represented as a conjugation of smaller cycles. This step dominates the computational time of the algorithm, and its order is O(VE 3 ) Horton s algorithm ....

Horton, J.D., A polynomial-time algorithm to find the shortest cycle basis of a graph, SIAM J. Comput., 16(2):358--366, 1987.

....Cycle bases of graphs have a variety of applications in science and engineering. For example, applications occur in structural flexibility analysis [9] electrical networks [3] and in chemical structure storage and retrieval systems [5] Brief surveys and extensive references can be found in [8, 7]. The set R of relevant cycles of a graph G is the union of its minimum cycle bases [12, 15] We define an equivalence relation interchangeability on R such that the cycles in a class W # P of the associated partition P can be expressed as a sum of a linearly independent set consisting of ....

....The length #(B) of a cycle basis B is the sum of the lengths of its generalized cycles: #(B) P C#B C . A minimum cycle basis M is a cycle basis with minimum length. The generalized the electronic journal of combinatorics 7 (2000) #R16 3 cycles in M are chord less cycles (see [8]) Hence we may consider cycles instead of generalized cycles from here on. For the sake of completeness we note that a minimum cycle basis is a cycle basis in which the longest cycle has the minimum possible length [2] Definition 1. 12] A cycle C is relevant if it cannot be represented as an ....

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J. D. Horton. A polynomial-time algorithm to find the shortest cycle basis of a graph. SIAM J. Comput., 16:359--366, 1987.

....and hence each of the 2 b Gamma1 subsets of B that contain the maximal element of B, is an elementary cycle. The Minimal Cycle Basis of a Secondary Structure is Unique Horton proposed an algorithm for finding a minimum cycle basis in polynomial time which is based on the following Proposition [8]. If C is a member of a minimum cycle base then for all for all vertices x in C there is an edge (y; z) 2 C such that C = P (x; y) Phi P (x; z) Phi (y; z) where P (a; b) denotes a shortest path from a to b. Remark. The converse is not true in general. Definition. An elementary cycle C is a ....

J. D. Horton. A polynomial-time algorithm to find the shortest cycle basis of a graph. SIAM J. Comput., 16:359--366, 1987.

....a minimum weight subgraph of G with a given cyclomatic number d is called the d cycle problem. The cyclomatic number d is considered as a constant. The d cycle problem is related to the problem of finding a minimum cycle basis, that is a minimum weight basis of the cycle space of G. Horton [6] obtained an O(nm 3 ) algorithm for finding a minimum cycle basis of a graph. Hartvigsen [5] extended Horton s technique to minimum shortest paths (of course, a cycle can be considered as a closed path) and to more general linear spaces. For any U V Theta V , he defined the U space, that is ....

Horton, J. D.: A polynomial time algorithm to find the shortest cycle basis of a graph. SIAM J. Comput., 16 (2) (1987) 358--366

....of a generalized cycle C is the number of its edges. The length #(B) of a cycle basis B is the sum of the lengths of its generalized cycles: #(B) # C#B C . A minimum cycle basis is a cycle basis with minimal length. Polynomial time algorithms for computing minimum cycle bases are known [6, 5]. If G is outerplanar, then its minimal cycle basis is unique [7] In general, of course, this is not true. In this contribution we show that almost all Halin graphs have the planar basis F as the unique minimum cycle basis, and we determine all alternative minimum cycle bases of the exceptional ....

Horton J. D. (1987). A polynomial-time algorithm to find the shortest cycle basis of a graph. SIAM J. Comput., 16:359--366.

....set R consists exactly of the relevant cycles [25] which cannot be represented as a # sum of shorter cycles. Vismara [29, 30] proposed an algorithm for computing R that works by first extracting so called prototypes from a set of short cycles similar to Horton s algorithm for finding a MCB [16]. The computation of the prototypes requires O( E 3 #(#) operations. The set R is then obtained by a backtracking procedure from the prototypes with O( V R ) operations. For some classes of graphs R grows exponentially with V , see [30] for an example. However, in 6 Table 1 ....

J. D. Horton. A polynomial-time algorithm to find the shortest cycle basis of a graph. SIAM J. Comput., 16:359--366, 1987.

....Cycle bases of graphs have a variety of applications in science and engineering. For example, applications occur in structural flexibility analysis [9] electrical networks [3] and in chemical structure storage and retrieval systems [5] Brief surveys and extensive references can be found in [8, 7]. The set R of relevant cycles of a graph G is the union of its minimum cycle bases [12, 15] We define an equivalence relation interchangeability on R such that the cycles in a class W # P of the associated partition P can be expressed as a sum of a linearly independent set consisting of ....

....The length #(B) of a cycle basis B is the sum of the lengths of its generalized cycles: #(B) # C#B C . A minimum cycle basis M is a cycle basis with minimum length. The generalized the electronic journal of combinatorics 7 (2000) #R16 3 cycles in M are chord less cycles (see [8]) Hence we may consider cycles instead of generalized cycles from here on. For the sake of completeness we note that a minimum cycle basis is a cycle basis in which the longest cycle has the minimum possible length [2] Definition 1. 12] A cycle C is relevant if it cannot be represented as an ....

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J. D. Horton. A polynomial-time algorithm to find the shortest cycle basis of a graph. SIAM J. Comput., 16:359--366, 1987.

....the partial relations to the problem of finding a set of fundamental cycles in a graph. Each fundamental cycle of the graph leads to one ordinary relation. The algorithm does not find the shortest fundamental cycles because of the enormous time and place requirements of such an algorithm (see [4], 8] Practical experiences (mainly with NFS for factoring integers) show that up to 200 partial relations are involved in composing one ordinary relation. This leads to much heavier systems of linear equations than in the original NFS algorithm. These heavier systems are more difficult to solve ....

J. D. Horton, A polynomial-time algorithm to find the shortest cycle basis of a graph, SIAM J. Comput. 16 (1987) , pp. 344 - 355

....the equations corresponding to the loops representing a magnetostatic problem. The basis functions are generated by first constructing a graph G whose nodes are elements of the discretization and whose edges represent connections between elements. We then find a spanning tree T and a cycle basis C [5] for G. The curl free basis functions are composed of pairs of rooftop functions that represent current flow across edges in T . Each loop in C gives rise to a divergence free basis function. Expanding the current J using the basis functions and testing against the same functions in the standard ....

J. D. Horton. A polynomial-time algorithm to find the shortest cycle basis of a graph. SIAM J. Comput., 16(2):358--366, Apr. 1987.

....edge (u; v) in T , and we use a pair of rooftop functions to represent a current flowing out of triangle u and into triangle v. For the divergence free basis functions, we must be able to represent the current flow in an arbitrary loop in the structure. To this end, we require a cycle basis C [4] for G. The most common way to construct a cycle basis is to use the spanning tree T . If we pick an arbitrary node and designate it as the root of T , then each edge (u; v) in G that is not in T defines a cycle as follows. We begin at the root, follow the unique path in T from the root to u, ....

....v back to the root. Unfortunately, this tends to produce large cycles, even if T has been constructed in a breadth first manner. This is undesirable for reasons that we will discuss in Subsection 4.2. We found that much shorter cycles can be obtained using the heuristic method proposed by Horton [4]. The types of graphs that arise from boundary meshes tend to be nearly planar. For such graphs, Horton s heuristic can be implemented so as to run in roughly linear time, and the bases that it produces are nearly minimal. We construct one divergence free basis function for each loop in the cycle ....

J. D. Horton. A polynomial-time algorithm to find the shortest cycle basis of a graph. SIAM J. Comput., 16(2):358--366, Apr. 1987.

....the graph is split apart into two pieces at the cut vertex, each part is embedded in the the plane separately, and then the pieces are put back together again. Because it was most important to select cycles with few incident edges, we applied Theorem 3.1 instead of Theorem 3.2. Horton s algorithm [10] with carefully chosen edge weights was used to select a cycle basis guaranteed to minimize the exponential component of the algorithm. 4 Computational Results A graph G is a topological obstruction if it has no degree two vertices and removing any edge from the graph results in a graph with ....

J. D. Horton. A polynomial time algorithm to find the shortest cycle basis of a graph. SIAM J. Comp., 16:358--366, 1987.

....vismara ensam.inra.fr Submited: October 10, 1996; Accepted: January 17, 1997. Abstract The perception of cyclic structures is a crucial step in the analysis of graphs. To describe the cycle vector space of a graph, a minimum cycle basis can be computed in polynomial time using an algorithm of [Horton, 1987]. But the set of cycles corresponding to a minimum basis is not always relevant for analyzing the cyclic structure of a graph. This restriction is due to the fact that a minimum cycle basis is generally not unique for a given graph. Therefore, the smallest canonical set of cycles which describes ....

....they are compared according to their lengths. The length of a cycle basis is the sum of the lengths of all cycles in the basis. This notion defines minimum fundamental cycle bases. Deo et al. 1982] have shown that finding a minimum fundamental cycle basis is an NP complete problem. Furthermore, [Horton, 1987] has presented a polynomial time algorithm to find a minimum cycle basis not necessarily associated with a spanning tree. Some applications require the perception of the cyclic parts of a graph. An obvious solution would be to list all elementary cycles. This method has two drawbacks: it cannot be ....

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J. D. Horton. A polynomial-time algorithm to find the shortest cycle basis of a graph. SIAM J. Comput., 16(2):358--366, 1987.

....2.1.3. Theorem 2.15 is a rephrasing of a result by Simonis [145] Finding a basis formed by minimum weight codewords seems to be a difficult problem. A related algorithmic problem of finding a basis with a minimal total weight of all vectors in it was studied by Chickering et al. 40] Horton [85]. Theorem 2.16 is due to Gelfand et al. 74] In an earlier related work [73] Gelfand and Dobrushin proved that there are codes for which the encoding circuit has size O(n log n) and depth O(log n) 2.2 Other models of noise We intend to take a brief look at codes correcting other types of ....

....evolution form the subject of matroid theory; see Welsh [165] Many problems that are difficult for general linear codes, are polynomial for cycle codes of graphs. Examples are decoding, see Ntafos and Hakimi [124] finding a basis of minimal total weight, see Chickering et al. 40] Horton [85], and computing the covering radius, see Frank [66] Theorem 4.4 is due to McLoughlin [122] The complexity of the Weight of error problem with preprocessing (P 9 ) was proved in Bruck and Naor [33] Their reduction is from K 3 . The proof that we give, following Lobstein [24, pp. 121 123] has ....

J. D. Horton, "A polynomial-time algorithm to find the shortest cycle basis of a graph," SIAM J. Comput., 16 (2) (1987), 358--366.

....can be decomposed in polynomial time into 1 sums, 2 sums, and 3 sums of graphic matroids, cographic matroids and the special ten element matroid R 10 . Truemper [Tru90]gives an algorithm which finds such a decomposition in cubic time. An algorithm to solve the MCB problem for graphs is given in [Hor87]. The Gomory Hu tree of [GH61]solves the MCB problem for cographic matroids. The main technical result of this paper is to showhow the minimal cycle 1 bases of a decomposition can be glued together to form a minimal cycle basis of the k sum. 2 Background The symmetric difference of two sets is ....

....Because B is a minimal cycle basis, w(B) w(B 0 ) Hence w(C) w(C j ) In other words, a cycle is not in the minimum cycle basis if and only if it can be written as the sum of smaller cycles. 3. 2 MCB in graphic matroids The algorithm to find the minimal cycle basis for a weighted graph in [Hor87] is based on the following: Lemma 3.2 Let C be a cycle in a minimal cycle basis of a graph G,and let x be a vertex of C. Then thereisanedge e 2 C, say e = fu# vg,such that C consists of a shortest path from u to x and a shortest path from v to x and the edge e. Therefore the following finds ....

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J. D. Horton. A polynomial-time algorithm to find the shortest cycle basis of a graph. SIAM Journal on Computing, 16(2):358-- 366, 1 1987.

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J. D. Horton. A polynomial-time algorithm to find a shortest cycle basis of a graph. SIAM Journal of Computing, 16:359--366, 1987.

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Horton, J.D.: A polynomial-time algorithm to find a shortest cycle basis of a graph. SIAM Journal of Computing 16 (1987) 359--366

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J.D. Horton. A polynomial-time algorithm to find the shortest cycle basis of a graph. SIAM Journal of Computing, 16(2):358--366, 1987.

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J.D.Horton. A polynomial-time algorithm to find the shortest cycle basis of a graph. SIAM Journal on Computing, 16(2):358--366, April 1987.

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J.D.Horton. A polynomial-time algorithm to find the shortest cycle basis of a graph. SIAM Journal on Computing, 16(2):358--366, April 1987.

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J. D. Horton. A polynomial-time algorithm to find the shortest cycle basis of a graph. SIAM J. Comput., 16:359--366, 1987.

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J. D. Horton. A polynomial-time algorithm to find the shortest cycle basis of a graph. SIAM J. Comput., 16:359--366, 1987.

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J. D. Horton. A polynomial-time algorithm to find the shortest cycle basis of a graph. SIAM J. Comput., 16:359--366, 1987. J. Leydold & P. F. Stadler: Minimal Cycle Bases of Outerplanar Graphs 12

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J. D. Horton. A polynomial-time algorithm to find the shortest cycle basis of a graph. SIAM J. Comput., 16:359--366, 1987.

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