A unified approach to domination problems on interval graphs

Gomathi, R. M.; Rangan, C. Pandu

doi:10.1016/0020-0190(88)90091-9

Cited by 107 publications

(67 citation statements)

References 7 publications

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“…The class of interval graphs is hereditary, that is, every induced subgraph of an interval graph G is also an interval graph. Ramalingam and Rangan [21] proposed a numbering of the vertices of an interval graph; they stated the following lemma. …”

Section: Structural Properties Of Interval Graphsmentioning

confidence: 99%

“…This numbering, which also results after sorting the intervals of the intersection model of an interval graph G on their right ends [1], can be obtained in O(|V (G)| + |E(G)|) time [21]. An ordering of the vertices according to this numbering is found to be quite useful in solving some graph-theoretic problems on interval graphs [1,21].…”

Section: |V (G)| Such That If I < J < K and Ik ∈ E(g) Then Jk ∈ mentioning

confidence: 99%

“…An ordering of the vertices according to this numbering is found to be quite useful in solving some graph-theoretic problems on interval graphs [1,21]. Throughout the paper, such an ordering is called a right-end ordering of G. Let u and v be two vertices of G, and let π be a right-end ordering of G; by u < π v we denote that u appears before v in π .…”

Section: |V (G)| Such That If I < J < K and Ik ∈ E(g) Then Jk ∈ mentioning

confidence: 99%

“…It has been shown that we can obtain the right-end ordering π of G, which results from numbering the intervals after sorting them on their right ends, in O(n + m) time [1,21].…”

Section: Time Complexitymentioning

confidence: 99%

See 3 more Smart Citations

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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Section: Structural Properties Of Interval Graphsmentioning

confidence: 99%

Section: |V (G)| Such That If I < J < K and Ik ∈ E(g) Then Jk ∈ mentioning

confidence: 99%

Section: |V (G)| Such That If I < J < K and Ik ∈ E(g) Then Jk ∈ mentioning

confidence: 99%

“…It has been shown that we can obtain the right-end ordering π of G, which results from numbering the intervals after sorting them on their right ends, in O(n + m) time [1,21].…”

Section: Time Complexitymentioning

confidence: 99%

See 2 more Smart Citations

The Longest Path Problem has a Polynomial Solution on Interval Graphs

2010

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show abstract

“…The class of interval graphs is hereditary, that is, every induced subgraph of an interval graph G is also an interval graph. Ramalingam and Rangan [18] proposed a numbering of the vertices of an interval graph; they stated the following lemma. [18]): The vertices of any interval graph G can be numbered with integers 1, 2, .…”

Section: Structural Properties Of Interval Graphsmentioning

confidence: 99%

The Longest Path Problem Is Polynomial on Interval Graphs

Ioannidou

Mertzios

Nikolopoulos

2009

Mathematical Foundations of Computer Science 2009

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Abstract. 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 in [20], 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 runs in O(n 4 ) time, where n is the number of vertices of the input graph, and bases on a dynamic programming approach.

show abstract

The Complexity of the Exact Weighted Independent Set Problem

Milanič

Monnot²

2008

Combinatorial Optimization and Theoretical Computer Science

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A unified approach to domination problems on interval graphs

Cited by 107 publications

References 7 publications

The Longest Path Problem has a Polynomial Solution on Interval Graphs

The Longest Path Problem has a Polynomial Solution on Interval Graphs

The Longest Path Problem Is Polynomial on Interval Graphs

The Complexity of the Exact Weighted Independent Set Problem

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