Linear-time algorithms for the Hamiltonian problems on distance-hereditary graphs

Hung, Ruo-Wei; Chang, Mau Song

doi:10.1016/j.tcs.2005.04.009

Cited by 32 publications

(20 citation statements)

References 42 publications

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“…It is sure that the Hamiltonian cycle problem is one of most well known NP-hard problem, and it is still NP-hard even for a chordal graph, and that an O( |V | + |E| ) time algorithm is known for distance hereditary graphs [17]. Here, we show that the Hamiltonian cycle problem can be solved in O( |V | ) time using the technique if a ptolemaic graph is given in the tree representation.…”

Section: Introductionmentioning

confidence: 87%

“…Especially, Bandelt and Mulder showed that any distance hereditary graph can be obtained from K 2 by a sequence of extensions called "adding a pendant vertex" and "splitting a vertex." Using the characterizations, many efficient algorithms have been found for distance hereditary graphs [6,2,5,21,17,7]. However, the recognition of distance hereditary graphs in linear time is not so simple; Hammer and Maffray's algorithm [14] fails in some cases, and Damiand, Habib, and Paul's algorithm [9] requires to build a cotree in linear time (see [9,Chapter 4] for further details), where the cotree can be constructed in linear time by using recent algorithm based on multisweep LBFS approach by Bretscher, Corneil, Habib, and Paul [4].…”

Section: Introductionmentioning

confidence: 99%

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Laminar Structure of Ptolemaic Graphs and Its Applications

Uehara

Uno

2005

Algorithms and Computation

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Ptolemaic graphs are graphs that satisfy the Ptolemaic inequality for any four vertices. The graph class coincides with the intersection of chordal graphs and distance hereditary graphs. The graph class can also be seen as a natural generalization of block graphs (and hence trees). In this paper, a new characterization of ptolemaic graphs is presented. It is a laminar structure of cliques, and leads us to a canonical tree representation. The tree representation gives a simple intersection model for ptolemaic graphs. The tree representation is constructed in linear time from a perfect elimination ordering obtained by the lexicographic breadth first search. Hence the recognition and the graph isomorphism for ptolemaic graphs can be solved in linear time. Using the tree representation, we also give an O(n) time algorithm for the Hamiltonian cycle problem. The Hamiltonian cycle problem is NP-hard for chordal graphs, and an O(n + m) time algorithm is known for distance hereditary graphs.

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Section: Introductionmentioning

confidence: 87%

Section: Introductionmentioning

confidence: 99%

Laminar Structure of Ptolemaic Graphs and Its Applications

Uehara

Uno

2005

Algorithms and Computation

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“…Recently, Hsieh et al [13] presented an O(n + m)-time sequential algorithm for the Hamiltonian problem on a distance-hereditary graph and also proposed a parallel implementation of their algorithm, which solves the problem in O(log n) time using O((n + m)/ log n) processors on a PRAM model. A unified approach to solving the Hamiltonian problems on distancehereditary graphs was presented in [14], while Hsieh [12] presented an efficient parallel strategy for the 2HP problem on the same class of graphs. Algorithms for the path cover problem on other classes of graphs were proposed in [2,15,22].…”

Section: Related Workmentioning

confidence: 99%

A linear‐time algorithm for the k‐fixed‐endpoint path cover problem on cographs

Asdre

Nikolopoulos

2007

Networks

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In this paper, we study a variant of the path cover problem, namely, the k -fixed-endpoint path cover problem. Given a graph G and a subset T of k vertices of V (G), a kfixed-endpoint path cover of G with respect to T is a set of vertex-disjoint paths P that covers the vertices of G such that the k vertices of T are all endpoints of the paths in P. The k -fixed-endpoint path cover problem is to find a k -fixed-endpoint path cover of G of minimum cardinality; note that, if T is empty, that is, k = 0, the stated problem coincides with the classical path cover problem. We show that the k -fixed-endpoint path cover problem can be solved in linear time on the class of cographs. More precisely, we first establish a lower bound on the size of a minimum k -fixed-endpoint path cover of a cograph and prove structural properties for the paths of such a path cover. Then, based on these properties, we describe an algorithm which, for a cograph G on n vertices and m edges, computes a minimum k -fixed-endpoint path cover of G in linear time, that is, in O(n + m) time. The proposed algorithm is simple, requires linear space, and also enables us to solve some path cover related problems, such as the 1HP and 2HP, on cographs within the same time and space complexity.

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“…Although the original Hamiltonian path problem is proven to be an NPC problem [2], the time complexity of finding Hamiltonian cycles can be reduced to polynomial time under certain conditions, such as finding Hamiltonian cycles in distance-hereditary graphs [3], or in co-comparability graphs [4], or in semi-complete multipartite digraphs [5], and so on.…”

Section: Introduction Backgroundmentioning

confidence: 99%

An Algorithm for Computing a Hamiltonian Cycle of Given Points in a Polygonal Region

Jia¹,

Jiang²

2018

dtcse

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In this paper, for a point set X in a simple polygon P, we present an algorithm to compute a Hamiltonian cycle which avoids the boundary of P and accesses all points of X. Computing a Hamiltonian cycle of given points in a simple polygon is a novel transformation of the general Hamiltonian problems and the problem of finding simple paths on given obstacles in a simple polygon. In our solution, we first construct the geodesic convex hull G of X inside P, and then find simple paths turn only at the points of X for two adjacent vertices in X along the boundary of G which are invisible to each other. After an original Hamiltonian cycle is found, we insert the remaining points to it. Finally, we present an O((n 2 +m)nlogm) time algorithm (n is the number of points in X and m is the number of P's vertices) to report a Hamiltonian cycle if it exists or report that no such cycle exists.

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Linear-time algorithms for the Hamiltonian problems on distance-hereditary graphs

Cited by 32 publications

References 42 publications

Laminar Structure of Ptolemaic Graphs and Its Applications

Laminar Structure of Ptolemaic Graphs and Its Applications

A linear‐time algorithm for the k‐fixed‐endpoint path cover problem on cographs

An Algorithm for Computing a Hamiltonian Cycle of Given Points in a Polygonal Region

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