J. E. Hopcroft and R. E. Tarjan. Isomorphism of planar graphs (working paper). In R. E. Miller and J. W. Thatcher, editors, Complexity of Computer Computations. Plenum Press, 1972.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of unit edges, and then all other edges are re directed to an appropriate node in the chain. Of the two main steps above, the second one is trivial to implement. The rst one is done by invoking some variant of the classical algorithm for testing tree isomorphism devised by Hopcroft and Tarjan [6] in connection with their linear planarity test (see, e.g. 1] Chapt. 3, pp. 84 86) Although this algorithm is linear and not prohibitively involved, we show here that there is a faster and more natural way to test isomorphism on subword trees. Combined with the simplicity of Step 2, this makes ....

J. E. Hopcroft and R. E. Tarjan, Isomorphism of Planar Graphs, in R. E. Miller and J. W. Thatcher (eds.) Complexity of Computer Computations, 131-150 Plenum Press (1972).

....the subgraph isomorphism problem when G and H are trees. Figure 1 gives an instance of this problem. Throughout this work n and k denote the number of vertices in G and H, respectively. When k = n we get the tree isomorphism problem, which has a linear time algorithm due to Hopcroft and Tarjan [13]. Polynomial algorithms for subtree isomorphism were first given by Matula [20] and by Edmonds (cf. 21] Faster algorithms, with worst case time complexity of O(k 1:5 n) were given by Matula [21] and Chung [4] In contrast, the subgraph isomorphism problem is NP complete when G is a tree and ....

J. E. Hopcroft and R. E. Tarjan. Isomorphism of planar graphs. In Raymond E. Miller and James W. Thatcher, Complexity of Computer Computations, Plenum Press, pages 131--152. 1972.

....paper we study subtree isomorphism, i.e. the subgraph isomorphism problem when G and H are trees. Throughout this work n and k denote the number of vertices in G and H , respectively. When k = n we get the tree isomorphism problem, which has a linear time algorithm due to Hopcroft and Tarjan [10]. In contrast, the problem is NP complete when G is a tree and H is a forest (subforest isomorphism [7] Polynomial algorithms for subtree isomorphism were first given by Matula [16] and by Edmonds (cf. 17] Faster algorithms, with worst case time complexity of O(k 1:5 n) were given ....

J. E. Hopcroft and R. E. Tarjan. Isomorphism of planar graphs. In Raymond E. Miller and James W. Thatcher, Complexity of Computer Computations, Plenum Press. 1972.

....adding some new ones; in particular, the data structures are new. email: fdititz, itai, rodehg cs.technion.ac.il 1 1 Introduction Free tree isomorphism is a well known problem which has many applications. A linear time algorithm to solve the problem has been developed by Hopcroft and Tarjan [2], who used it to devise a linear planarity test. Several other linear algorithms were suggested [1, pp.196 199] 3] 4] The common approach is as follows. First, free tree isomorphism is reduced to the rooted case. The root is set canonically at the central vertex or at the new vertex connected ....

....1: Bad tree for algorithm. b) root label = 1 label = 2 label = 2 D 2 Tvertices = fu 2 g label = 2 Tvertices = fu 4 g u 1 u 4 (a) 1 = 1 = 1 = 1 = 1 = 2 = 2 = 1 u 2 = 1 u 3 = 2 = 2 = 2 T Tvertices = fu 1 ; u 3 g Tvertices = f g Figure 2: a) The tree T , level [2] = u 1 ; u 2 ; u 3 ; u 4 ) b) The tree D 2 . Zemlyachenko August 3, 4 down the tree. We now scan all the children of level [h] by nondecreasing index. When considering a vertex u, we move its parent v from d, the D vertex to which it was associated, to d 0 , the child of d whose label is ....

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J. E. Hopcroft and R. E. Tarjan. Isomorphism of planar graphs. In R. E. Miller and J. W. Thatcher, editors, Complexity of Computer Computations, pages 131--150. Plenum Press, 1972.

....In particular, Immerman and Lander [8] and Lindell [9] proved that the counting extension IFP C of IFP, introduced by Immerman [7] captures P on the class of trees. We study the problem of capturing P on the class of planar graphs. From the fact (essentially due to Hopcroft and Tarjan [4]) that there is a P canonization algorithm for planar graphs it follows that there is a logic that captures P on planar graphs. However, this logic is quite abstract and unsatisfying from a logicians perspective. We prove that the ILV Theorem extends to 3 connected planar graphs, and that the ....

J. E. Hopcroft and R. Tarjan. Isomorphism of planar graphs (working paper). In R. E. Miller and J. W. Thatcher, editors, Complexity of Computer Computations. Plenum Press, 1972.

....color and either both adjacent, or both not adjacent to the other chosen pair. Thus Player II can always preserve the partial isomorphism. 2 1. 8 Vertex Refinement Corresponds to C 2 It turns out that the expressive power of C 2 is characterized by the well known method of vertex refinement (see [2, 16]) Let G = hV; E; C 1 ; C r i 1. Describing Graphs: a First Order Approach to Graph Canonization 17 be a colored graph in which every vertex statisfies exactly one color relation. Let f : V f1 : ng be given by f(v) i iff v 2 C i . We then define f 0 , the refinement of f as ....

....[26] It follows from his results that: Corollary 1.8.6 For all d, and sufficiently large n, C 3 characterizes more than 1 Gamma O[1=n] of the regular graphs of degree d on n vertices. 1. 9 Equivalence and Canonization Algorithms The stable coloring of a graph is computable in O[jEj log n] steps [16]. We present the algorithm for completeness. Algorithm 1.9.1 1. Place indices 1; r of initial color classes on list L. 2. While L 6= do begin 3. For each vertex v adjacent to some color classes in L, record how many neighbors of each such color class v has. 4. Sort these records to ....

John E. Hopcroft and Robert Tarjan, "Isomorphism of Planar Graphs," in Complexity of Computer Computations, R. Miller and J.W Thatcher, eds., (1972), Plenum Press, 131-152.

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J. E. Hopcroft and R. E. Tarjan. Isomorphism of planar graphs (working paper). In R. E. Miller and J. W. Thatcher, editors, Complexity of Computer Computations. Plenum Press, 1972.

No context found.

J. E. Hopcroft and R. E. Tarjan. Isomorphism of planar graphs. In R. E. Miller and J. W. Thatcher, editors, Complexity of Computer Computations, pages 131--150. Plenum Press, 1972.

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J. E. Hopcroft and R. Tarjan. Isomorphism of planar graphs (working paper). In R. E. Miller and J. W. Thatcher, editors, Complexity of Computer Computations. Plenum Press, 1972.

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