Computational Complexity of Geodetic Set

Atıcı, Mustafa

doi:10.1080/00207160210954

Cited by 46 publications

(32 citation statements)

References 4 publications

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“…1 we can see, on the one hand, that {v 1 , v 4 } does not induce a maximal geodetic closure because I({v 1 …”

Section: Maximal Geodetic Closuresmentioning

confidence: 97%

“…The complexity of the problem of finding the geodetic number has been studied in [1] where Atici has shown that the corresponding GEODETIC SET decision problem, namely "given a nontrivial connected graph G = (V, E) and an integer k |V|, is there a set S ⊂ V with |S| = k and such that I…”

Section: Introductionmentioning

confidence: 99%

“…The geodetic number was first introduced in 1990 (see [4]) and is further studied in [1][2][3][5][6][7][8] and [9]. Several classes of graphs and their geodetic number are presented in [8].…”

Section: Introductionmentioning

confidence: 99%

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On pitfalls in computing the geodetic number of a graph

Hansen

Omme

2006

Optimization Letters

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Given a simple connected graph G = (V, E) the geodetic closure I[S] ⊂ V of a subset S of V is the union of all sets of nodes lying on some geodesic (or shortest path) joining a pair of nodes v k , v l ∈ S. The geodetic number, denoted by g(G), is the smallest cardinality of a node set S * such that I[S * ] = V. In "The geodetic number of a graph", [Harary et al. in Math. Comput. Model. 17:89-95, 1993] propose an incorrect algorithm to find the geodetic number of a graph G. We provide counterexamples and show why the proposed approach must fail. We then develop a 0-1 integer programming model to find the geodetic number. Computational results are given.

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“…1 we can see, on the one hand, that {v 1 , v 4 } does not induce a maximal geodetic closure because I({v 1 …”

Section: Maximal Geodetic Closuresmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

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On pitfalls in computing the geodetic number of a graph

Hansen

Omme

2006

Optimization Letters

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“…A set S of vertices is a geodetic set if I [S] = V, and the minimum cardinality of a geodetic set is the geodetic number g (G). A geodetic set of cardinality g(G) is called a g-set of G. The geodetic number of a graph was introduced in [1,3] and further studied in [2,4,9]. It was shown in [1] that determining the geodetic number of a graph is an NP-hard problem.…”

Section: Introductionmentioning

confidence: 99%

“…A geodetic set of cardinality g(G) is called a g-set of G. The geodetic number of a graph was introduced in [1,3] and further studied in [2,4,9]. It was shown in [1] that determining the geodetic number of a graph is an NP-hard problem. A set S of vertices is an edge geodetic set of a graph G if each edge of G lies on a geodesic between two vertices from S, and the minimum cardinality of an edge geodetic set is the edge geodetic number eg (G).…”

Section: Introductionmentioning

confidence: 99%

The 2-edge geodetic number and graph operations

Santhakumaran

Chandran

2012

Arab. J. Math.

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For a connected graph G = (V, E) of order n ≥ 2, a set S ⊆ V is a 2-edge geodetic set of G if each edge e ∈ E − E(S) lies on a u-v geodesic with d(u, v) = 2 for some vertices u and v in S. The minimum cardinality of a 2-edge geodetic set in G is the 2-edge geodetic number of G, denoted by eg 2 (G). It is proved that for any connected graph G, β 1 (G) ≤ eg 2 (G), where β 1 (G) is the matching number of G. It is shown that every pair a, b of integers with 2 ≤ a ≤ b is realizable as the matching number and 2-edge geodetic number, respectively, of some connected graph. We determine bounds for the 2-edge geodetic number of Cartesian product of graphs. Also we determine the 2-edge geodetic number of certain classes of Cartesian product graphs. The 2-edge geodetic number of join of two graphs is obtained in terms of the 2-edge geodetic number of the factor graphs.

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