# The forcing geodetic number of a graph

Discussiones Mathematicae Graph Theory (1999)

- Volume: 19, Issue: 1, page 45-58
- ISSN: 2083-5892

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topGary Chartrand, and Ping Zhang. "The forcing geodetic number of a graph." Discussiones Mathematicae Graph Theory 19.1 (1999): 45-58. <http://eudml.org/doc/270610>.

@article{GaryChartrand1999,

abstract = {For two vertices u and v of a graph G, the set I(u, v) consists of all vertices lying on some u-v geodesic in G. If S is a set of vertices of G, then I(S) is the union of all sets I(u,v) for u, v ∈ S. A set S is a geodetic set if I(S) = V(G). A minimum geodetic set is a geodetic set of minimum cardinality and this cardinality is the geodetic number g(G). A subset T of a minimum geodetic set S is called a forcing subset for S if S is the unique minimum geodetic set containing T. The forcing geodetic number $f_G(S)$ of S is the minimum cardinality among the forcing subsets of S, and the forcing geodetic number f(G) of G is the minimum forcing geodetic number among all minimum geodetic sets of G. The forcing geodetic numbers of several classes of graphs are determined. For every graph G, f(G) ≤ g(G). It is shown that for all integers a, b with 0 ≤ a ≤ b, a connected graph G such that f(G) = a and g(G) = b exists if and only if (a,b) ∉ (1,1),(2,2).},

author = {Gary Chartrand, Ping Zhang},

journal = {Discussiones Mathematicae Graph Theory},

keywords = {geodetic set; geodetic number; forcing geodetic number; geodetic closure; shortest paths},

language = {eng},

number = {1},

pages = {45-58},

title = {The forcing geodetic number of a graph},

url = {http://eudml.org/doc/270610},

volume = {19},

year = {1999},

}

TY - JOUR

AU - Gary Chartrand

AU - Ping Zhang

TI - The forcing geodetic number of a graph

JO - Discussiones Mathematicae Graph Theory

PY - 1999

VL - 19

IS - 1

SP - 45

EP - 58

AB - For two vertices u and v of a graph G, the set I(u, v) consists of all vertices lying on some u-v geodesic in G. If S is a set of vertices of G, then I(S) is the union of all sets I(u,v) for u, v ∈ S. A set S is a geodetic set if I(S) = V(G). A minimum geodetic set is a geodetic set of minimum cardinality and this cardinality is the geodetic number g(G). A subset T of a minimum geodetic set S is called a forcing subset for S if S is the unique minimum geodetic set containing T. The forcing geodetic number $f_G(S)$ of S is the minimum cardinality among the forcing subsets of S, and the forcing geodetic number f(G) of G is the minimum forcing geodetic number among all minimum geodetic sets of G. The forcing geodetic numbers of several classes of graphs are determined. For every graph G, f(G) ≤ g(G). It is shown that for all integers a, b with 0 ≤ a ≤ b, a connected graph G such that f(G) = a and g(G) = b exists if and only if (a,b) ∉ (1,1),(2,2).

LA - eng

KW - geodetic set; geodetic number; forcing geodetic number; geodetic closure; shortest paths

UR - http://eudml.org/doc/270610

ER -

## References

top- [1] G. Chartrand, F. Harary and P. Zhang, The geodetic number of a graph, Networks (to appear). Zbl0987.05047
- [2] G. Chartrand, F. Harary, and P. Zhang, On the hull number of a graph, Ars Combin. (to appear). Zbl1064.05049

## Citations in EuDML Documents

top- A.P. Santhakumaran, J. John, On the forcing geodetic and forcing steiner numbers of a graph
- A.P. Santhakumaran, J. John, The forcing steiner number of a graph
- Gary Chartrand, Ping Zhang, The forcing dimension of a graph
- Gary Chartrand, Ping Zhang, On graphs with a unique minimum hull set
- Gary Chartrand, Frank Harary, Ping Zhang, Geodetic sets in graphs

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