In this paper we consider a k-center location problem, which is based on the dominating set problem. The computation of the minimum dominating sets of a graph is used as a basic step for the determination of k-centers of the graph. We first study reachable sets [2], [5], link vectors [2] of the vertices and then introduce two binary operations ⋁ and ⋀ for finding the dominating sets. These are used as tools for designing an algorithm to find the k-centers. We construct G λ-graphs [2], [4] and then we investigate their dominating sets. Some required results for developing algorithms are also proved. Using these concepts we present an algorithm to find all k-centers of a graph. We show that the domination based k-center problem is NP complete.

background or methods

Distance in graphs

Domination based algorithm to k-center problem

The known relation between the standard radius and diameter holds for graphs, but not for digraphs. We show that no upper estimation is possible for digraphs. We also give some remarks on distances, which are either metric or non-metric.

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Note on the relation between radius and diameter of a graph

The impact of the apex node on routing inside a pyramid structure

In this paper a new concept D-centro dominating set in graphs is introduced and graphs are characterized with some results. A subset S⊂V (G) of a connected graph G is said to be D-centro dominating set of G, if for every v ∈V −S, there exists a vertex u in S such that D(u,v) = Rad(G). The minimum cardinality of the D-centro dominating set is called D-centro domination number, denoted by DCγ(G). The D-centro dominating set with cardinality DCγ(G) is called DCγ -set of G. Some bounds for the D-centro domination number are determined. An important realization result on D-centro domination number is proved that for any integers a and b with 2≤ a≤ b, there exists a connected graph G such that DCγ(G) = a and DC(G) = b.

/pdf/d-centro-dominating-sets-in-graphs-28qyt8fzie.pdf

$D$-centro dominating sets in graphs

Given a connected simple graph G and S ⊆ V (G), the geodetic closure IG[S] of S is the set of all vertices lying on some u-v geodesic where u and v are in S. In this paper, select vertices of G sequentially as follows: select a vertex v1 and let S1 = {v1}. Select a vertex v2 6= v1 and let S2 = {v1, v2}, then determine IG[S2]. If IG[S2] 6= V (G), then successively select a vertex vi / ∈ IG[Si−1] and let Si = {v1, v2, . . . , vi} for i = 1, 2, . . . , k. Then determine IG[Si]. A subset S of V (G) is called a weakly connected closed geodetic set of G if the selection of vertex vk in the given manner yields IG[Sk] = V (G) and 〈S〉w is connected, where 〈S〉w = 〈N [S], Ew〉 with Ew consists of edges uv ∈ E(G) such that u ∈ S or v ∈ S. The minimum cardinality of weakly connected closed geodetic set is called the weakly connected closed geodetic number wcgn(G) of G. In this paper, the weakly connected closed geodetic sets of some common graphs and graphs resulting from the join of graphs are characterized. Also, the weakly connected closed geodetic numbers of these graphs are determined. This research is funded through the DOST-ASTHRDP. 258 Rachel M. Patangan, Imelda S. Aniversario and Rosalio G. Artes, Jr.

Distance in graphs

Citations

Domination based algorithm to k-center problem

Note on the relation between radius and diameter of a graph

The impact of the apex node on routing inside a pyramid structure

$D$-centro dominating sets in graphs

Weakly connected closed geodetic numbers of graphs

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