The hamiltonian cycle problem is linear-time solvable for 4-connected planar graphs

Chiba, Norishige; Nishizeki, Takao

doi:10.1016/0196-6774(89)90012-6

Cited by 80 publications

(47 citation statements)

References 8 publications

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“…Finding the separating triangles and making the graphs 4-connected takes linear time [3]. A Hamiltonian cycle in a 4-connected planar graphs can be found in linear time [4].…”

Section: Making the Graphs 4-connectedmentioning

confidence: 99%

“…Let G be the subgraph of G induced on vertices {1, 2, 3, 4, 5}, and G be the subgraph of G induced on vertices {2, 6, 7, 8, 9}. Since G is 3-connected fixing the outer-face fixes an embedding for G. With the given outer-face of G, the path P contains two crossings: one involving (2,4), and the other one involving (6,8). Graph G has six faces and unless we change the outer-face of G such that it contains the edge (1, 3) or (3,5), the edge (2, 4) is involved in a crossing in the path.…”

Section: Theoremmentioning

confidence: 99%

“…First we find a hamiltonian cycle H 1 of G 1 and a hamiltonian cycle H 2 of G 2 . We can do this in linear time using the algorithm of [4]. Starting at a random vertex in H 1 we traverse its vertices, assigning increasing x-coordinates to each vertex visited.…”

Section: Simultaneous Embedding With Few Bendsmentioning

confidence: 99%

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Simultaneous Embedding of Planar Graphs with Few Bends

Erten

Kobourov

2005

Graph Drawing

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Abstract. We present an O(n) time algorithm for simultaneous embedding of pairs of planar graphs on the O(n 2 ) × O(n 2 ) grid, with at most three bends per edge, where n is the number of vertices. For the case when the input graphs are both trees, only one bend per edge is required. We also describe an O(n) time algorithm for simultaneous embedding with fixed-edges for tree-path pairs on the O(n) × O(n 2 ) grid with at most one bend per tree-edge and no bends along path edges.

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“…Finding the separating triangles and making the graphs 4-connected takes linear time [3]. A Hamiltonian cycle in a 4-connected planar graphs can be found in linear time [4].…”

Section: Making the Graphs 4-connectedmentioning

confidence: 99%

Section: Theoremmentioning

confidence: 99%

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Simultaneous Embedding of Planar Graphs with Few Bends

Erten

Kobourov

2005

Graph Drawing

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“…Testing all possible edges e = (v, w) on the outer face, we could request a hamiltonian path, which is a well-known NP-complete problem even on planar graphs. On the other hand, we know of a linear-time algorithm to find external hamiltonian cycles by Chiba and Nishizeki [5], if the graph is four-connected. Since the graphs we consider are triangulated, the problematic cases appear if there are separating triangles, namely cycles of length 3 which do not circumscribe single faces.…”

Section: The General Casementioning

confidence: 99%

Embedding Vertices at Points: Few Bends Suffice for Planar Graphs

Kaufmann

Wiese

1999

Graph Drawing

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Abstract. The existing literature gives efficient algorithms for mapping trees or less restrictively outerplanar graphs on a given set of points in a plane, so that the edges are drawn planar and as straight lines. We relax the latter requirement and allow very few bends on each edge while considering general plane graphs. Our results show two algorithms for mapping four-connected plane graphs with at most one bend per edge and for mapping general plane graphs with at most two bends per edge. Furthermore we give a point set, where for arbitrary plane graphs it is NP-complete to decide whether there is an mapping such that each edge has at most one bend.

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“…If G is 4-connected we can find a Hamiltonian cycle and hence a (nearly) perfect matching in linear time [CN89]. If G is not 4-connected the basic idea is to find a matching in every block separately and combine them to a matching in G. Let G 1 , .…”

Section: The 4-block Tree Is a Pathmentioning

confidence: 99%

Computing large matchings fast

Rutter

Wolff²

2010

ACM Trans. Algorithms

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In this paper we present algorithms for computing large matchings in 3-regular graphs, graphs with maximum degree 3, and 3-connected planar graphs. The algorithms give a guarantee on the size of the computed matching and take linear or slightly superlinear time. Thus they are faster than the best-known algorithm for computing maximum matchings in general graphs, which runs in O( √ nm) time, where n denotes the number of vertices and m the number of edges of the given graph. For the classes of 3-regular graphs and graphs with maximum degree 3 the bounds we achieve are known to be best possible.We also investigate graphs with block trees of bounded degree, where the d-block tree is the adjacency graph of the d-connected components of the given graph. In 3-regular graphs and 3-connected planar graphs with bounded-degree 2-and 4-block trees, respectively, we show how to compute maximum matchings in slightly superlinear time.

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The hamiltonian cycle problem is linear-time solvable for 4-connected planar graphs

Cited by 80 publications

References 8 publications

Simultaneous Embedding of Planar Graphs with Few Bends

Simultaneous Embedding of Planar Graphs with Few Bends

Embedding Vertices at Points: Few Bends Suffice for Planar Graphs

Computing large matchings fast

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