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On the forcing geodetic and forcing steiner numbers of a graph

100%

Santhakumaran A., John J.

Discussiones Mathematicae Graph Theory

2011

tom 31

nr 4

611-624

For a connected graph G = (V,E), a set W ⊆ V is called a Steiner set of G if every vertex of G is contained in a Steiner W-tree of G. The Steiner number s(G) of G is the minimum cardinality of its Steiner sets and any Steiner set of cardinality s(G) is a minimum Steiner set of G. For a minimum Steiner set W of G, a subset T ⊆ W is called a forcing subset for W if W is the unique minimum Steiner set containing T. A forcing subset for W of minimum cardinality is a minimum forcing subset of W. The forcing Steiner number of W, denoted by fₛ(W), is the cardinality of a minimum forcing subset of W. The forcing Steiner number of G, denoted by fₛ(G), is fₛ(G) = min{fₛ(W)}, where the minimum is taken over all minimum Steiner sets W in G. The geodetic number g(G) and the forcing geodetic number f(G) of a graph G are defined in [2]. It is proved in [6] that there is no relationship between the geodetic number and the Steiner number of a graph so that there is no relationship between the forcing geodetic number and the forcing Steiner number of a graph. We give realization results for various possibilities of these four parameters.

The forcing steiner number of a graph

100%

Santhakumaran A., John J.

Discussiones Mathematicae Graph Theory

2011

tom 31

nr 1

171-181

For a connected graph G = (V,E), a set W ⊆ V is called a Steiner set of G if every vertex of G is contained in a Steiner W-tree of G. The Steiner number s(G) of G is the minimum cardinality of its Steiner sets and any Steiner set of cardinality s(G) is a minimum Steiner set of G. For a minimum Steiner set W of G, a subset T ⊆ W is called a forcing subset for W if W is the unique minimum Steiner set containing T. A forcing subset for W of minimum cardinality is a minimum forcing subset of W. The forcing Steiner number of W, denoted by fₛ(W), is the cardinality of a minimum forcing subset of W. The forcing Steiner number of G, denoted by fₛ(G), is fₛ(G) = min{fₛ(W)}, where the minimum is taken over all minimum Steiner sets W in G. Some general properties satisfied by this concept are studied. The forcing Steiner numbers of certain classes of graphs are determined. It is shown for every pair a, b of integers with 0 ≤ a < b, b ≥ 2, there exists a connected graph G such that fₛ(G) = a and s(G) = b.

Ograniczanie wyników

2 Discussiones Mathematicae Graph Theory

2 John J.

2 Santhakumaran A.

2 2011

Wyniki wyszukiwania

On the forcing geodetic and forcing steiner numbers of a graph

The forcing steiner number of a graph