For a vertex v of a connected graph G and a subset S of V(G), the distance between v and S is d(v,S) = mind(v,x)|x ∈ S. For an ordered k-partition Π = S₁,S₂,...,Sₖ of V(G), the representation of v with respect to Π is the k-vector r(v|Π) = (d(v,S₁), d(v,S₂),..., d(v,Sₖ)). The k-partition Π is a resolving partition if the k-vectors r(v|Π), v ∈ V(G), are distinct. The minimum k for which there is a resolving k-partition of V(G) is the partition dimension pd(G) of G. A resolving partition Π = S₁,S₂,...,Sₖ of V(G) is connected if each subgraph induced by (1 ≤ i ≤ k) is connected in G. The minimum k for which there is a connected resolving k-partition of V(G) is the connected partition dimension cpd(G) of G. Thus 2 ≤ pd (G) ≤ cpd(G) ≤ n for every connected graph G of order n ≥ 2. The connected partition dimensions of several classes of well-known graphs are determined. It is shown that for every pair a, b of integers with 3 ≤ a ≤ b ≤ 2a-1, there is a connected graph G having pd(G) = a and cpd(G) = b. Connected graphs of order n ≥ 3 having connected partition dimension 2, n, or n-1 are characterized.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1177,
author = {Varaporn Saenpholphat and Ping Zhang},
title = {Connected partition dimensions of graphs},
journal = {Discussiones Mathematicae Graph Theory},
volume = {22},
year = {2002},
pages = {305-323},
zbl = {1027.05037},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1177}
}
Varaporn Saenpholphat; Ping Zhang. Connected partition dimensions of graphs. Discussiones Mathematicae Graph Theory, Tome 22 (2002) pp. 305-323. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1177/
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