Deferred-query: An efficient approach for some problems on interval graphs

Chang, Mau Song; Peng, Sheng‐Lung; Liaw, Jenn-Liang

doi:10.1002/(sici)1097-0037(199908)34:1<1::aid-net1>3.0.co;2-c

Cited by 17 publications

(23 citation statements)

References 28 publications

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“…In the same paper, they ask whether the time bound for these two problems can be improved to O(n) time if a so-called sorted interval representation is given. Chang, Peng and Liaw [7] answered this question in the affirmative. They showed that this even holds for Path Cover.…”

Section: Testing For Hamilton Cycles and Hamilton Pathsmentioning

confidence: 99%

Linear-Time Algorithms for Scattering Number and Hamilton-Connectivity of Interval Graphs

Broersma

Fiala

Golovach

et al. 2013

Graph-Theoretic Concepts in Computer Science

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Abstract.We show that for all k ≤ −1 an interval graph is −(k + 1)-Hamilton-connected if and only if its scattering number is at most k. We also give an O(n + m) time algorithm for computing the scattering number of an interval graph with n vertices and m edges, which improves the O(n 3 ) time bound of Kratsch, Kloks and Müller. As a consequence of our two results the maximum k for which an interval graph is kHamilton-connected can be computed in O(n + m) time.

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Section: Testing For Hamilton Cycles and Hamilton Pathsmentioning

confidence: 99%

Linear-Time Algorithms for Scattering Number and Hamilton-Connectivity of Interval Graphs

Broersma

Fiala

Golovach

et al. 2013

Graph-Theoretic Concepts in Computer Science

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“…They have important properties, and admit polynomial time solutions for several problems that are NP-complete on general graphs (see e.g. [1,5,15,18]). Moreover, interval graphs have received a lot of attention due to their applicability to DNA physical mapping problems [14], and find many applications in several fields and disciplines such as genetics, molecular biology, scheduling, VLSI circuit design, archeology and psychology [15].…”

Section: Structural Properties Of Interval Graphsmentioning

confidence: 99%

“…6, H (i, x) = H (4,5), (4,14), and H (x + 1, j − 1) = H (6,14); then A(H (4, 5)) = {u 4 , u 5 }, C (H (4, 14)) = {u 6 , u 9 }, and C(H (6,14)) = {u 9 }, and thus V (H 1 ) = A(H (4, 5)) ∪ C (H (4, 14)) \ C (H (6, 14)) = {u 4 , u 5 , u 6 }).…”

Section: Next Show That In This Case Algorithm Lp_on_h Computes (U mentioning

confidence: 99%

“…On the other hand, there are several classes of graphs on which the Hamiltonian path problem admits polynomial time solutions; these classes include proper interval graphs [3], interval graphs [1,5,8], circular-arc graphs [8], biconvex graphs [2], and cocomparability graphs [6]. Thus, if someone is interested in investigating the tractability of the longest path problem, it makes sense to focus on the classes of graphs for which the Hamiltonian path problem is polynomial.…”

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confidence: 94%

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The Longest Path Problem has a Polynomial Solution on Interval Graphs

2010

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The longest path problem is the problem of finding a path of maximum length in a graph. Polynomial solutions for this problem are known only for small classes of graphs, while it is NP-hard on general graphs, as it is a generalization of the Hamiltonian path problem. Motivated by the work of Uehara and Uno (Proc. of the 15th Annual International Symp. on Algorithms and Computation (ISAAC), LNCS, vol. 3341, pp. 871-883, 2004), where they left the longest path problem open for the class of interval graphs, in this paper we show that the problem can be solved in polynomial time on interval graphs. The proposed algorithm uses a dynamic programming approach and runs in O(n 4 ) time, where n is the number of vertices of the input graph.

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“…The same holds true for bipartite graphs [27], split graphs [17], circle graphs [10], undirected path graphs [2] and grid graphs [26]. However, polynomial time algorithms exist for the Hamiltonian cycle or Hamiltonian path problem on some special classes of graphs, such as interval graphs [1,7], permutation graphs [13,36], cocomparability graphs [12,14], Ptolemaic graphs [8] and distance-hereditary graphs [19,22,25].…”

Section: Introductionmentioning

confidence: 99%

Solving the path cover problem on circular-arc graphs by using an approximation algorithm

Hung

Chang

2006

Discrete Applied Mathematics

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A path cover of a graph G = (V , E) is a family of vertex-disjoint paths that covers all vertices in V. Given a graph G, the path cover problem is to find a path cover of minimum cardinality. This paper presents a simple O(n)-time approximation algorithm for the path cover problem on circulararc graphs given a set of n arcs with endpoints sorted. The cardinality of the path cover found by the approximation algorithm is at most one more than the optimal one. By using the result, we reduce the path cover problem on circular-arc graphs to the Hamiltonian cycle and Hamiltonian path problems on the same class of graphs in O(n) time. Hence the complexity of the path cover problem on circulararc graphs is the same as those of the Hamiltonian cycle and Hamiltonian path problems on circular-arc graphs.

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Deferred-query: An efficient approach for some problems on interval graphs

Cited by 17 publications

References 28 publications

Linear-Time Algorithms for Scattering Number and Hamilton-Connectivity of Interval Graphs

Linear-Time Algorithms for Scattering Number and Hamilton-Connectivity of Interval Graphs

The Longest Path Problem has a Polynomial Solution on Interval Graphs

Solving the path cover problem on circular-arc graphs by using an approximation algorithm

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