Geodetic Sets in Graphs

Brešar, Boštjan; Kovše, Matjaž; Tepeh, Aleksandra

doi:10.1007/978-0-8176-4789-6_8

Cited by 24 publications

(19 citation statements)

References 61 publications

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“…monophonic) set if every vertex of G lies in a geodetic (resp. monophonic) interval between two vertices from S. (See the survey on geodetic sets in graphs [5].) The geodetic (monophonic) number g(G) (mn(G)) of a graph G is the minimum cardinality of a geodetic (monophonic) set in G.…”

Section: Some Invariants Arising From Toll Convexitymentioning

confidence: 99%

“…In the resulting toll convexity the interval graphs are precisely the graphs which are convex geometry. Then we focus on two standard invariants, in relation with this newly introduced type of graph convexity (for a study of these invariants in relation with geodesic convexity see the survey [5]). Finally, we describe the structure of toll convex sets in three graph products, which has also been done for some other types of convexities [3,27].…”

Section: Introductionmentioning

confidence: 99%

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Toll convexity

Alcón

Brešar

Gologranc

et al. 2015

European Journal of Combinatorics

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a b s t r a c tA walk W between two non-adjacent vertices in a graph G is called tolled if the first vertex of W is among vertices from W adjacent only to the second vertex of W , and the last vertex of W is among vertices from W adjacent only to the second-last vertex of W . In the resulting interval convexity, a set S ⊂ V (G) is toll convex if for any two non-adjacent vertices x, y ∈ S any vertex in a tolled walk between x and y is also in S. The main result of the paper is that a graph is a convex geometry (i.e. satisfies the Minkowski-Krein-Milman property stating that any convex subset is the convex hull of its extreme vertices) with respect to toll convexity if and only if it is an interval graph. Furthermore, some well-known types of invariants are studied with respect to toll convexity, and toll convex sets in three standard graph products are completely described.

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Section: Some Invariants Arising From Toll Convexitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Toll convexity

Alcón

Brešar

Gologranc

et al. 2015

European Journal of Combinatorics

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“…Observe that the first case simplifies to (n, m) ∈ {(3, 3), (3,4), (4, 4), (4, 5), (5,5), (6,6)} and that the "otherwise" case appears if and only if m ≥ n ≥ 7 and m ≤ 3 + n−3 2 . Note that the second case is indeed a known result from [19, Corollary 2.3], but here we present a different proof for it.…”

Section: Complete Bipartite Graphsmentioning

confidence: 99%

“…. The only possibilities are (n, m) ∈ {(3, 3), (3, 4), (4, 4), (4, 5), (5,5), (6,6)}. For all of them we can easily check that the optimal value is m.…”

Section: Complete Bipartite Graphsmentioning

confidence: 99%

Strong Geodetic Number of Complete Bipartite Graphs, Crown Graphs and Hypercubes

Gledel

Iršič

2019

Bull. Malays. Math. Sci. Soc.

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The strong geodetic number, sg(G), of a graph G is the smallest number of vertices such that by fixing one geodesic between each pair of selected vertices, all vertices of the graph are covered. In this paper, the study of the strong geodetic number of complete bipartite graphs is continued. The formula for sg(K n,m ) is given, as well as a formula for the crown graphs S 0 n . Bounds on sg(Q n ) are also discussed.

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“…A set S of vertices in a graph G is called the geodetic set of G if for every vertex x ∈ V (G) there exist u, v ∈ S such that x ∈ I(u, v). The geodetic number g(G) of a graph G is the least size of a set of vertices S such that any vertex from G lies on a u, v-geodesic, where u, v ∈ S. We refer to [5,8] for surveys on geodetic sets in graphs.…”

mentioning

confidence: 99%

On the remoteness function in median graphs

Balakrishnan

Brešar

Changat

et al. 2009

Discrete Applied Mathematics

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a b s t r a c tA profile on a graph G is any nonempty multiset whose elements are vertices from G.The corresponding remoteness function associates to each vertex x ∈ V (G) the sum of distances from x to the vertices in the profile. Starting from some nice and useful properties of the remoteness function in hypercubes, the remoteness function is studied in arbitrary median graphs with respect to their isometric embeddings in hypercubes. In particular, a relation between the vertices in a median graph G whose remoteness function is maximum (antimedian set of G) with the antimedian set of the host hypercube is found. While for odd profiles the antimedian set is an independent set that lies in the strict boundary of a median graph, there exist median graphs in which special even profiles yield a constant remoteness function. We characterize such median graphs in two ways: as the graphs whose periphery transversal number is 2, and as the graphs with the geodetic number equal to 2. Finally, we present an algorithm that, given a graph G on n vertices and m edges, decides in O(m log n) time whether G is a median graph with geodetic number 2.

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Geodetic Sets in Graphs

Cited by 24 publications

References 61 publications

Toll convexity

Toll convexity

Strong Geodetic Number of Complete Bipartite Graphs, Crown Graphs and Hypercubes

On the remoteness function in median graphs

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